Abstract Algebra II: Introduction to Rings PDF
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Kwame Nkrumah University of Science and Technology
2023
B. O. Bainson
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This document is an introduction to rings in abstract algebra. It covers ring definitions, properties, and examples, along with exercises and proofs. The work is suitable for undergraduate mathematics students and from the Kwame Nkrumah University of Science and Technology.
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ABSTRACT ALGEBRA II Introduction to Rings B. O. Bainson (PhD) Kwame Nkrumah University of Science and Technology July 11, 2023 B. O. Bainso...
ABSTRACT ALGEBRA II Introduction to Rings B. O. Bainson (PhD) Kwame Nkrumah University of Science and Technology July 11, 2023 B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 1 / 98 1 Introduction 2 Preliminaries Definition Basic properties of rings Special elements in a ring Special classes of rings 3 Subrings 4 Ring homomorphims 5 Ideals 6 Quotient rings 7 Isomorphisms Theorems 8 Polynomial rings 9 Euclidean Domains 10 Unique Factorization Domains B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 2 / 98 Introduction Ring theory is a fundamental area in pure mathematics. Rings are studied owing to many applications of mathematics in Physics, Chemistry, Biology, Finance, Economics, Engineering and many more are built on vector spaces which are special case of ring theory. Many algorithmic processes in Computer Science rely chiefly on intrinsic properties of fields which is a special of rings. It is therefore imperative to investigate the many intrinsic beauty of rings. Ring theory is predominantly among the favourite subjects of many undergraduate students. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 3 / 98 In algebra, we are most algebraic objects of study are sets together with some operation(s). Under these operation(s), we investigate properties that arise. So far we have considered objects such as I Z I Zp I Q I R I Z[x] I Mn (R) I C. under some basic arithmetic operations B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 4 / 98 Introduction Complete the table below Property Z Q R C Zn Zp Mn (R) Closure under addition Addition is associative Existence of additive identity Every element has an additive inverse Addition is commutative Closure under multiplication Multiplication is associative Multiplication is commutative Multiplication is distributive over addition Existence of multiplicative identity Every element has a multiplicative inverse There are no zero divisors B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 5 / 98 Remark 1 Recall that objects satisfying first four properties a classified as groups. If in addition property 5 holds, then it is an abelian group. 2 It is evident that some abelian groups such as Z also admit multiplication operation which interact ”nicely” with the addition operation. 3 We call these objects rings. 4 Analogous to using permutations to study group theory, we will motivate our study of rings with Z. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 6 / 98 Definition(Ring) A ring is a set R together with two operations called addition (+) and multiplication (·) such that the following axioms hold: 1 The set R is closed under the addition operation. 2 Addition is associative. 3 Addition is commutative. 4 The set R contains an additive identity element (0R ). 5 Every element has an additive inverse in R. 6 Multiplication is associative. 7 Multiplication is distributive over addition. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 7 / 98 Axiomatic Description of Rings Remark The axioms 1-5 shows that (R, +) is an abelian group. Thus we can describe a ring as an abelian group equipped with a multiplication operation satisfying axioms 6-7. Immediate and probably boring but important ring is R = {0}. By a routine check, all the axioms hold. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 8 / 98 Examples of Rings 1 The set Mn (R) together with the usual matrix addition and multiplication is a ring. In general what can you say about Mn (Z), Mn (Q), Mn (R) and Mn (C)? 2 The cyclic group Zn is a ring. 3 The cyclic group nZ is a ring. 4 The real Hamilton Quaternions H consisting of numbers of the form a + bi + cj + dk where a, b, c, d ∈ R with the usual addition and multiplication is a ring. 5 The set F of all functions f : R → R is an abelian group under the addition operation (f + g )(x) = f (x) + g (x). If we define multiplication on F by (fg )(x) = f (x)g (x). Then F together with these operations is a ring. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 9 / 98 Exercise √ √ 1 Show that the set Z[ 2] = {a + b 2 : a, b ∈ Z} together with the usual addition and multiplication is a ring. 2 Show that the Gaussian integers Z[i] = {a + bi : a, b ∈ Z} together with the usual addition and multiplication is a ring. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 10 / 98 The following results are easily deducible from the axioms of rings. Theorem If R is a ring with additive identity 0, then for any a, b ∈ R, we have 1 0a = 0 = a0. 2 a(−b) = (−a)b = −(ab). 3 (−a)(−b) = ab. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 11 / 98 Proof. 1. 0a = (0 + 0)a = 0a + 0a. Subtracting 0a from both sides we get 0 = 0a as desired. 2. Note that −(ab) is the additive inverse of ab. Thus to show property 2. It suffices to show that a(−b) + (ab) = 0. We note that a(−b) + ab = a(−b + b) = a0 = 0 Similarly, (−a)b + ab = (−a + a)b = 0b = 0 3.Exercise B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 12 / 98 Special elements in a ring Remark 1 The additive identity denoted by 0R in ring R is called a zero element. 2 A multiplicative identity is not a defining property of rings. Hence a ring may or may not have a multiplicative identity. 3 If a ring R has a multiplicative identity denoted by 1R , we call 1R the identity (or unity) and the ring a ring with identity (or unity). 4 Consequently, in an arbitrary ring, if ac = cb and a 6= 0, we can not conclude that c = b. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 13 / 98 Properties Theorem Let R be a ring. Then the zero element is unique. Proof. Let z1 and z2 be distinct zero elements in the ring R. By definition, a + z1 = a and b + z2 = b for all a, b ∈ R. Replacing a with z2 and b with z1 we have z2 = z2 + z1 = z1. Thus z1 = z2. Theorem Let R be a ring with unity 1R.Then 1R is unique. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 14 / 98 Definition Let R be a ring and a ∈ R. We say that a is; 1 a unit if a has a multiplicative inverse. 2 a zero divisor if a 6= 0 and there is a nonzero element b ∈ R such that ab = ba = 0; 3 nilpotent if ak = 0 for some k ∈ N; 4 idempotent if a2 = a. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 15 / 98 Identify the identity, units, zero divisors, nilpotents and idempotents in 1 Z. 2 Zn for n = 4, 5. 3 H. 4 M2 (Z2 ) 5 Z×Z √ 6 Q[ 2] B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 16 / 98 Proposition Let R be a ring with identity. Then the set R × of units in R is an abelian group under the multiplication defined on R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 17 / 98 Proposition Let R be a ring with identity. Then the set R × of units in R is an abelian group under the multiplication defined on R. Proof. Let R be a ring with identity 1 and R × be the set of units in R. Then 1 ∈ R × is the multiplicative identity in R ×.Also, since multiplication in R is associative and commutative, they hold for R ×. It suffices to show closure under multiplication and inverses. Let a, b ∈ R ×. Then a−1 and b −1 are in R. But ab ∈ R × since (ab)(b −1 a−1 ) = 1. Furthermore, R × is closed under inversion since a−1 ∈ R × for all a ∈ R ×. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 17 / 98 ( ) P 1 Let G be a group. Show that ZG = ng g : ng ∈ Z with g ∈G P P P operations below is a ring ng g + ag g = (ng + ag )g g ∈G g ∈G g ∈G ! P P P and ng g ah h = (ng ah )gh g ∈G h∈G g ∈G √ 2 Determine if the set A = {a + b 2 : a, b ∈ 2Z} together with the usual addition and multiplication is a ring. 3 Define the multiplications · and · on Z by a · b = 0 and a ◦ b = 1. Determine if (Z, +, ·) and (Z, +, ◦) are rings. 4 Show whether or not the set of fourth complex roots of unity forms a ring under usual addition and mulitplication of complex numbers. 5 Is the set of all numbers a/b in Q where gcd(a, b) = 1 and b is odd a ring? B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 18 / 98 Remark V It is noted that a ring may or may not all the special elements described earlier. For example the ring 2Z does not have an identity. Similarly, the set of all functions f : R → R with compact support is a ring with no identity. V Cosequently, we classify rings based on these elements as follows. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 19 / 98 Definition( Commutative rings) Let R be a ring. Then R is said to be commutative if the multiplication operation on R is commutative. Definition (Domain) A commutative ring with identity is called an integral domain, or simply a domain if has no zero divisors. Remark V An immediate example in mind will be Z. V We emphasize that the absence of zero divisors leads to the cancellation property in such rings (including Z). V We formally describe this cancellation property as follows. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 20 / 98 Theorem Suppose R is an integral domain and a, b, c ∈ R with a 6= 0. If ab = ac, then b = c. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 21 / 98 Theorem Suppose R is an integral domain and a, b, c ∈ R with a 6= 0. If ab = ac, then b = c. Proof. Suppose that R is a domain and let a, b, c ∈ R with a 6= 0 and ab = ac. Then ab − ac = 0 and so a(b − c) = 0. However, since a 6= 0, we must have that b − c = 0 and hence b = c as desired. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 21 / 98 Now we can classify the following rings. Example Which of the following are domains? 1 Z 2 Z5 3 R 4 Q 5 C 6 Z6 7 Q[x], 8 Z2 9 M2 (Z2 ) B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 22 / 98 Definition (Division ring) A ring R with identity is called a division ring (or skew field) if every nonzero element has a multiplicative inverse. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 23 / 98 Definition (Division ring) A ring R with identity is called a division ring (or skew field) if every nonzero element has a multiplicative inverse. Example 1 Z5 2 R 3 Q 4 C B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 23 / 98 Definition(Field) A commutative ring with unity in which every non-zero element is a unit is called a field. Alternatively, a field is a commutative division ring. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 24 / 98 Definition(Field) A commutative ring with unity in which every non-zero element is a unit is called a field. Alternatively, a field is a commutative division ring. Example Which of the following are fields? 1 Z 2 Q 3 R 4 C 5 Z5 6 Z[i] B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 24 / 98 Theorem Every field F is an integral domain. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 25 / 98 Theorem Every field F is an integral domain. Proof. Let F be a field and let a, b ∈ F with a 6= 0. Suppose ab = 0. We have that a−1 (ab) = 0. But 0 = a−1 (ab) = (a−1 a)b = b. Therefore a is not a zero divisor B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 25 / 98 Theorem Every finite integral domain is a field. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 26 / 98 Theorem Every finite integral domain is a field. Proof. Let R be a finite integral domain. By definition, R has no zero dvisors. Let a be a nonzero element. Then the map φ : R → R defined by x 7→ ax is in particular injective since ax = ay implies x = y by the cancellation property. Since R is finite, this map is also surjective and so there is some b ∈ R such that ab = 1. Thus a is a unit (has a multiplicative inverse) in R. Since a was an arbitrary element, R is a field. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 26 / 98 Definition( Characteristic of a ring) Let be a ring R. The smallest positive integer n such that n · a = 0 for all a ∈ R is called the characteristic of R. If no such n exist, then R is said to have characteristic zero. Example Find the characteristics of the following rings. 1 Zp 2 Q 3 R 4 C B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 27 / 98 Subrings By now it is clear that some rings naturally are contained in others. For example 2Z ⊂ Z and M2 (2Z) ⊂ M2 (Z). Similarly, R ⊂ C. Consequently, it is necessary to have terminologies that perfectly describe these situations. An important note here is given the bigger ring R, the subset S of R is also a ring with respect to the given operations on R. Formally, we have the following terminology. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 28 / 98 Definition (Subring) Let (R, +, ·) be a ring. A subset S of R is called a subring if (S, +, ·) is a ring. Remark Since S is a subset of R, most of the ring axioms will hold by the restriction of the operations to S. It therefore suffices to check for 1 S is an abelian group under addition. 2 Closure under multiplication. We summarize the check in the following tests. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 29 / 98 Subring Test 1 A subset S of a ring R is a subring if the following axioms hold for all a, b, c ∈ S: 1 0R ∈ S. 2 a + b ∈ S. 3 −a ∈ S. 4 ab ∈ S. Subring Test 2 A nonempty subset S of a ring R is a subring if the following axioms hold for all a, b, c ∈ S: 1 a−b ∈S 2 ab ∈ S. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 30 / 98 Subrings It is immediate that Example Z ⊂ Q ⊂ R ⊂ C. Z[i] ⊂ C. Is 2Z is a subring of Z. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 31 / 98 Subrings It is immediate that Example Z ⊂ Q ⊂ R ⊂ C. Z[i] ⊂ C. Is 2Z is a subring of Z. Exercise Determine if S is a subring of R if 1 S = Z and R = Z[x]. 2 S = {0, 2, 4} and R = Z6. 3 S = {0, 2, 4, 6, 8} and R = Z10. 4 S = 6Z and R = 2Z. 5 S = Z2 and R = Z. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 31 / 98 Theorem Let S be a subring of R. Let 0S and 0R be the zero elements of the rings S and R, respectively. Then 0S = 0R. Furthermore, if a ∈ S, then (−a)S = (−a)R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 32 / 98 Theorem Let S be a subring of R. Let 0S and 0R be the zero elements of the rings S and R, respectively. Then 0S = 0R. Furthermore, if a ∈ S, then (−a)S = (−a)R. Proof. Let x ∈ S. Then we have x + 0S = x and x + 0R = x. But in R, x + 0S = x + 0R. Thus 0S = 0R. Furthermore, in S, we have that x + (−x)S = 0S and x + (−x)R = 0R. Since 0S = 0R , we have that x + (−x)S = x + (−x)R. Hence (−x)S = (−x)R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 32 / 98 Ring homomorphisms Note In group theory, a significantly advantageous means of investigating the structure of groups is by means of group homomorphism. Recall that homomorphisms are maps between objects of same kind that preserves the nature (structure) of the objects. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 33 / 98 Heinstein, “Images of homomorphisms is well likened to photographs”. Herein, just like taking a photograph of some object, although some 3D information may be lost, say the back of the object, the image may contain sufficiently important and readily accessible information due its“simplicity” to characterize the object. It is clear that the photographic image of an object does have adequate structural similarity with the object. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 34 / 98 Summary Group homomorphisms are maps between groups that preserve the group structure. Consequently, we describe ring homomorphisms as structure preserving maps between rings. In particular, a V ring homomorphism must preserve summation V ring homomorphism must preserve multiplication. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 35 / 98 Definition(Ring homomorphisms) Let (R, +, ·) and (Ω, ⊕, ) be rings. A ring homomorphism is a mapping φ : R → Ω such that 1 φ(a + b) = φ(a) ⊕ φ(b). 2 φ(a · b) = φ(a) φ(b). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 36 / 98 Example We can identify C in M2 (R) by the correspondence 1 0 0 1 1 7→ and i 7→. 0 1 −1 0 Consequently, z = a + ib will correspond to 1 0 0 1 a b a +b = 0 1 −1 0 −b a The sum and product of (a + ib) and(c + id) respectively identify a+c b+d a b c d = + −(b + d) a + c −b a −d c ac − bd ad + bc a b c d = −(ad + bc) ac − bd −b a −d c B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 37 / 98 Example Determine if the following maps are ring homomorphisms. 1 θ : C → C where a + ib 7→ a − ib. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 38 / 98 Example Determine if the following maps are ring homomorphisms. 1 θ : C → C where a + ib 7→ a − ib. 2 φ : Z → Zn where φ(a) is the remainder of a modulo n. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 38 / 98 Example Determine if the following maps are ring homomorphisms. 1 θ : C → C where a + ib 7→ a − ib. 2 φ : Z → Zn where φ(a) is the remainder of a modulo n. 3 θ : Z → 2Z where θ(a) = 2a. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 38 / 98 Example Determine if the following maps are ring homomorphisms. 1 θ : C → C where a + ib 7→ a − ib. 2 φ : Z → Zn where φ(a) is the remainder of a modulo n. 3 θ : Z → 2Z where θ(a) = 2a. 4 θ : Q[x] → Q where θ(f ) = f (2). 5 θ : Z → Z × Z where z 7→ (z, z). 6 For any n ∈ Z, the map ϕn : Z → Z defined by ϕn (x) = nx. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 38 / 98 Example Determine if the following maps are ring homomorphisms. 1 θ : C → C where a + ib 7→ a − ib. 2 φ : Z → Zn where φ(a) is the remainder of a modulo n. 3 θ : Z → 2Z where θ(a) = 2a. 4 θ : Q[x] → Q where θ(f ) = f (2). 5 θ : Z → Z × Z where z 7→ (z, z). 6 For any n ∈ Z, the map ϕn : Z → Z defined by ϕn (x) = nx. If a ring homomorphism φ : R → Ω is both injective and surjective, then is is said to be a ring isomorphism and we write R ∼ = Ω. Herein, we can find φ−1 : Ω → R such that φφ−1 and φ−1 φ are the identity mappings on Ω and R respectively. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 38 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then 1 φ(0R ) = 0Ω. 2 φ(−a) = −φ(a) and φ(a − b) = φ(a) − φ(b). 3 If R has an identity and φ is onto and Ω 6= {0}. Then φ(a) is a unit in Ω. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 39 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then 1 φ(0R ) = 0Ω. 2 φ(−a) = −φ(a) and φ(a − b) = φ(a) − φ(b). 3 If R has an identity and φ is onto and Ω 6= {0}. Then φ(a) is a unit in Ω. Proof. 1 We have that 0R = 0R + 0R Thus φ(0R ) = φ(0R + 0R ) = φ(0R ) + φ(0R ). Subtracting φ(0R ) from both sides gives 0Ω = 0R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 39 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then 1 φ(0R ) = 0Ω. 2 φ(−a) = −φ(a) and φ(a − b) = φ(a) − φ(b). 3 If R has an identity and φ is onto and Ω 6= {0}. Then φ(a) is a unit in Ω. Proof. 1 We have that 0R = 0R + 0R Thus φ(0R ) = φ(0R + 0R ) = φ(0R ) + φ(0R ). Subtracting φ(0R ) from both sides gives 0Ω = 0R. 2 0Ω = φ(0R ) = φ(a + (−a)) = φ(a) + φ(−a). Adding −φ(a) to both sides give −φ(a) = φ(−a). The remaining is left as an exercise. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 39 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then φ(R) is a subring of Ω. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 40 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then φ(R) is a subring of Ω. Proof. It suffices to check that φ(R) is closed under subtraction and multiplication. Let a, b ∈ φ(R), then a = φ(x) and b = φ(y ) for some x, y ∈ R. Then a − b = φ(x) − φ(y ) = φ(x − y ) ∈ φ(R). Similarly, ab = φ(x)φ(y ) = φ(xy ) ∈ φ(R). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 40 / 98 Definition Let φ : R → Ω be a ring homomorphism. The kernel of φ written ker(φ) is the set ker(φ) = {r ∈ R : φ(r ) = 0Ω } B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 41 / 98 Definition Let φ : R → Ω be a ring homomorphism. The kernel of φ written ker(φ) is the set ker(φ) = {r ∈ R : φ(r ) = 0Ω } Examples Find the kernel of the following maps. 1 φ : Z × Z → Z defined by (a, b) 7→ a. 2 φ : Q[x] → Q defined by φ(f (x)) = f (2). 3 φ : C → C defined by z 7→ z̄. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 41 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then kerφ is a subring of R. Furthermore, if α ∈ kerφ then r α, αr ∈ kerφ for every r ∈ R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 42 / 98 Theorem Let φ : R → Ω be a ring homomorphism. Then kerφ is a subring of R. Furthermore, if α ∈ kerφ then r α, αr ∈ kerφ for every r ∈ R. Proof. It suffices to check that ker(R) is closed under subtraction and multiplication. Let a, b ∈ kerφ. Then φ(a) = 0 and φ(b) = 0. Now φ(a − b) = φ(a) − φ(b) = 0 − 0 = 0. Hence (a − b) ∈ kerφ. Similarly, φ(ab) = φ(a)φ(b) = 0 · 0 = 0. Hence ab ∈ kerφ. Now, suppose α ∈ kerφ. Then φ(r α) = φ(r )φ(α) = φ(r ) · 0 = 0 and φ(αr ) = 0 · φ(r ). Hence αr , r α ∈ kerφ. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 42 / 98 Summary Kernels of ring homomorphisms are abelian subgroups. That is closed under ”sums” and ”differences” Kernels of ring homomorphisms are cloased under multiplication by any element of the ring. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 43 / 98 Summary Kernels of ring homomorphisms are abelian subgroups. That is closed under ”sums” and ”differences” Kernels of ring homomorphisms are cloased under multiplication by any element of the ring. Note Recall that kernels of group homomorphisms are special subgroups (normal subgroups). Normal subgroups are special since the set of cosets of normal subgroups can be given a group structure under appropriate definition. Kernels of ring homomorphisms are similarly special and so we consider as follows. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 43 / 98 Ideal Definition(Ideal) Let R be a ring. A subset I of R is called a left (right) ideal of R if 1 I is a subring of R. 2 I is closed under left (right) multiplication by elements of R. If I is both a left ideal and a right ideal, then I is called an ideal of R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 44 / 98 Ideal test Let R be a ring. A subset S ⊆ R is an ideal of R if 1 0R ∈ S. 2 a + b ∈ S for all a.b ∈ S. 3 −a ∈ S for all a ∈ S. 4 r ∈ R and s ∈ S, then rs ∈ S. Immediate examples are The subring I = {0} of any ring R is called the trivial ideal sometimes denoted by 0. he subring I = R of any ring R is an ideal. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 45 / 98 Examples A. Which of the following are ideals of the given rings; 1 The subset nZ of Z. 2 The set I of matrices in M2 (Z) with zero determinants. 3 The set of all multiples of any polynomial p(x) ∈ Z[x]. 4 Let a ∈ Z. The set of all polynomials p(x) ∈ Z[x] such that p(a) = 0 B. Find all the ideals in Z6. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 46 / 98 Remark 1 The ideal generated by a subset S of a ring R is the smallest ideal containing all elements of S, and is denoted hSi. 2 The elements of S are called generators of the ideal. 3 If S = {s}, then the ideal is called principal.. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 47 / 98 Remark 1 The ideal generated by a subset S of a ring R is the smallest ideal containing all elements of S, and is denoted hSi. 2 The elements of S are called generators of the ideal. 3 If S = {s}, then the ideal is called principal.. Example Which of the following ideals are principal? 1 h12, 9i of Z. 2 hx 2 + 3x, x 3 − x 2 + 3x − 3i of Q[x]. 3 h2, xi of Z[x]. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 47 / 98 Note 1 In domains such as Z and Q[x] every ideal is principal and so we call such domains are called principal ideal domains. 2 A commutative ring in which every ideal can be generated by a finite number of elements is called a Noetherian ring. Examples include R[x1 , · · · , xn ] where R = Z, Q, R or C. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 48 / 98 Proposition Let {Ia }a∈A be a family of ideals in R, indexed by a set A. Then ∩a∈A Ia is an ideal of R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 49 / 98 Proposition Let {Ia }a∈A be a family of ideals in R, indexed by a set A. Then ∩a∈A Ia is an ideal of R. Theorem Let R be a commutative ring with identity. Suppose r ∈ R, then < r > ⊆ < s > if and only if r = st for some t ∈ R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 49 / 98 Proposition Let {Ia }a∈A be a family of ideals in R, indexed by a set A. Then ∩a∈A Ia is an ideal of R. Theorem Let R be a commutative ring with identity. Suppose r ∈ R, then < r > ⊆ < s > if and only if r = st for some t ∈ R. Why study ideals?? B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 49 / 98 Quotient rings Note Let I be an ideal of a ring R. Then I is a subgroup of R and since R is an abelian group, I is normal in R. Thus, we can form the quotient group R/I using the operation (I + a) (I + b) = I + (a + b). V Our goal is to form a ring with R/I using the addition operation and an appropriate multiplication operation. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 50 / 98 Lemma(Properties of cosets) Let I be an ideal of a commutative ring R. Suppose a, b ∈ R. Then 1 if (I + a) ⊆ (I + b), then (I + a) = (I + b). 2 if (I + a) ∩ (I + b) 6= 0, then (I + a) = (I + b). 3 if (I + a) = (I + b) if and only if (a − b) ∈ I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 51 / 98 Lemma(Properties of cosets) Let I be an ideal of a commutative ring R. Suppose a, b ∈ R. Then 1 if (I + a) ⊆ (I + b), then (I + a) = (I + b). 2 if (I + a) ∩ (I + b) 6= 0, then (I + a) = (I + b). 3 if (I + a) = (I + b) if and only if (a − b) ∈ I. Proof. 1 It suffices to show that (I + b) ⊆ (I + a). Suppose (I + a) ⊆ (I + b). We have that a = 0 + a ∈ (I + a) ⊆ (I + b). Thus there is x ∈ I such that a = x + b. Consequently, b = −x + a ∈ (I + a). Now let y ∈ I , then y + b = (y − x) + a ∈ (I + a) and so (I + b) ⊆ (I + a). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 51 / 98 Lemma(Properties of cosets) Let I be an ideal of a commutative ring R. Suppose a, b ∈ R. Then 1 if (I + a) ⊆ (I + b), then (I + a) = (I + b). 2 if (I + a) ∩ (I + b) 6= 0, then (I + a) = (I + b). 3 if (I + a) = (I + b) if and only if (a − b) ∈ I. Proof. 1 It suffices to show that (I + b) ⊆ (I + a). Suppose (I + a) ⊆ (I + b). We have that a = 0 + a ∈ (I + a) ⊆ (I + b). Thus there is x ∈ I such that a = x + b. Consequently, b = −x + a ∈ (I + a). Now let y ∈ I , then y + b = (y − x) + a ∈ (I + a) and so (I + b) ⊆ (I + a). 2 Let (I + a) ∩ (I + b) 6= 0. Suppose c ∈ (I + a) ∩ (I + b). Then c ∈ (I + a) and so (I + c) ⊆ (I + a). By (1) (I + c) = (I + a). Similarly (I + a) = (I + c) ⊆ (I + b). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 51 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). Well-defined check. We show that the operations are well defined. Suppose (I + a) = (I + b) and (I + r ) = (I + s). For the addition, we show that (I + a) (I + r ) = (I + b) (I + s). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). Well-defined check. We show that the operations are well defined. Suppose (I + a) = (I + b) and (I + r ) = (I + s). For the addition, we show that (I + a) (I + r ) = (I + b) (I + s).That is, I + (a + r ) = I + (b + s). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). Well-defined check. We show that the operations are well defined. Suppose (I + a) = (I + b) and (I + r ) = (I + s). For the addition, we show that (I + a) (I + r ) = (I + b) (I + s).That is, I + (a + r ) = I + (b + s).By property 3 of above, we must show that (a + r ) − (b + s) ∈ I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). Well-defined check. We show that the operations are well defined. Suppose (I + a) = (I + b) and (I + r ) = (I + s). For the addition, we show that (I + a) (I + r ) = (I + b) (I + s).That is, I + (a + r ) = I + (b + s).By property 3 of above, we must show that (a + r ) − (b + s) ∈ I.But (a + r ) − (b + s) = (a − b) + (r − s) ∈ I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). Well-defined check. We show that the operations are well defined. Suppose (I + a) = (I + b) and (I + r ) = (I + s). For the addition, we show that (I + a) (I + r ) = (I + b) (I + s).That is, I + (a + r ) = I + (b + s).By property 3 of above, we must show that (a + r ) − (b + s) ∈ I.But (a + r ) − (b + s) = (a − b) + (r − s) ∈ I.For multiplication, we show that (I + a)(I + r ) = (I + b)(I + s). That is, I + (ar ) = I + (bs). Alternatively that ar − bs ∈ I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Definition Define the operations of addition and multiplication, respectively on R/I by (I + a) (I + b) = I + (a + b). (I + a)(I + b) = I + (ab). Well-defined check. We show that the operations are well defined. Suppose (I + a) = (I + b) and (I + r ) = (I + s). For the addition, we show that (I + a) (I + r ) = (I + b) (I + s).That is, I + (a + r ) = I + (b + s).By property 3 of above, we must show that (a + r ) − (b + s) ∈ I.But (a + r ) − (b + s) = (a − b) + (r − s) ∈ I.For multiplication, we show that (I + a)(I + r ) = (I + b)(I + s). That is, I + (ar ) = I + (bs). Alternatively that ar − bs ∈ I.However, ar − bs = ar − br + br − bs = (a − b)r + b(r − s) ∈ I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 52 / 98 Theorem Let I be an ideal of a ring R. Then the set of cosets R/I is a ring with the operations of addition and multiplication defined by (a + I ) (b + I ) = (a + b) + I. (a + I )(b + I ) = (ab) + I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 53 / 98 Theorem Let I be an ideal of a ring R. Then the set of cosets R/I is a ring with the operations of addition and multiplication defined by (a + I ) (b + I ) = (a + b) + I. (a + I )(b + I ) = (ab) + I. Definition(Quotient ring) Let R be a ring and I and ideal. Then the ring R/I is called the quotient ring of R by I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 53 / 98 Examples List (Describe) the elements of the following quotient rings. 1 Z/4Z 2 2Z/6Z 3 R[x]/hx 2 + 1i. 4 Z[x]/hx 2 − 2i. 5 Q[x]/x 2 − 2. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 54 / 98 Theorem[Canonical Projection] Let R be a ring and I and ideal. Then the mapping π : R → R/I defined by r 7→ I + r is a ring homomorphism from R onto R/I with kernel I. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 55 / 98 Theorem[Canonical Projection] Let R be a ring and I and ideal. Then the mapping π : R → R/I defined by r 7→ I + r is a ring homomorphism from R onto R/I with kernel I. Proof. Let a, b ∈ R. Then π(ab) = I + (ab) = (I + a)(I + b) = π(a)π(b). Also, π(a + b) = I + (a + b) = (I + a) + (I + a) = π(a) + π(b). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 55 / 98 Theorem[Canonical Projection] Let R be a ring and I and ideal. Then the mapping π : R → R/I defined by r 7→ I + r is a ring homomorphism from R onto R/I with kernel I. Proof. Let a, b ∈ R. Then π(ab) = I + (ab) = (I + a)(I + b) = π(a)π(b). Also, π(a + b) = I + (a + b) = (I + a) + (I + a) = π(a) + π(b).Now, since elements of R/I are of the form I + a where a ∈ R, and since π(a) = I + a by definition, the map is onto. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 55 / 98 Theorem[Canonical Projection] Let R be a ring and I and ideal. Then the mapping π : R → R/I defined by r 7→ I + r is a ring homomorphism from R onto R/I with kernel I. Proof. Let a, b ∈ R. Then π(ab) = I + (ab) = (I + a)(I + b) = π(a)π(b). Also, π(a + b) = I + (a + b) = (I + a) + (I + a) = π(a) + π(b).Now, since elements of R/I are of the form I + a where a ∈ R, and since π(a) = I + a by definition, the map is onto. Let a ∈ kerπ. Then πa = I + 0R = I + a. Thus a = (a − 0R ) ∈ I. Thus kerπ ⊆ I. Suppose i ∈ I , then π(i) = I + i = I + 0R and so i ∈ kerπ. Thus I ⊆ kerπ. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 55 / 98 Isomorphisms Theorems Theorem Let φ : R → S be a ring homomorphism. Then ker(φ) is an ideal of R. Proof. Exercise. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 56 / 98 Here are some isomorphism theorems analogous to that of groups. Theorem (First Isomorphism Theorem) Let φ : R → S be a surjective ring homomorphism and let π be the canonical projection map. Then there is a map ν : R/ker(φ) → S such that ν · π = φ. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Here are some isomorphism theorems analogous to that of groups. Theorem (First Isomorphism Theorem) Let φ : R → S be a surjective ring homomorphism and let π be the canonical projection map. Then there is a map ν : R/ker(φ) → S such that ν · π = φ. Proof. We seek to show that the diagram φ R / SO ν π $ R/ker(φ) B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Proof. We seek to show that the diagram φ R / SO ν π $ R/ker(φ) Define the map ν by ν(ker(φ) + r ) = φ(r ). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Proof. We seek to show that the diagram φ R / SO ν π $ R/ker(φ) Define the map ν by ν(ker(φ) + r ) = φ(r ).This is irrespective of the choice of coset representative. Since if ker(φ) + r = ker(φ) + s, then r − s ∈ ker(φ) so that φ(r ) = φ(s). Now check that ν is a ring homomorphism. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Proof. We seek to show that the diagram φ R / SO ν π $ R/ker(φ) Define the map ν by ν(ker(φ) + r ) = φ(r ).This is irrespective of the choice of coset representative. Since if ker(φ) + r = ker(φ) + s, then r − s ∈ ker(φ) so that φ(r ) = φ(s). Now check that ν is a ring homomorphism. ν((ker(φ) + x)(ker(φ) + y )) = ν(ker(φ) + xy ) = φ(xy ) = φ(x)φ(y ) = ν(ker(φ) + x)ν(ker(φ) + y ) B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Proof. We seek to show that the diagram φ R / SO ν π $ R/ker(φ) Define the map ν by ν(ker(φ) + r ) = φ(r ).This is irrespective of the choice of coset representative. Since if ker(φ) + r = ker(φ) + s, then r − s ∈ ker(φ) so that φ(r ) = φ(s). Now check that ν is a ring homomorphism. ν((ker(φ) + x) + (ker(φ) + y )) = ν(ker(φ) + (x + y )) = φ(x + y ) = φ(x) + φ(y ) = ν(ker(φ) + x) + ν(ker(φ) + y ) B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Proof. We seek to show that the diagram φ R / SO ν π $ R/ker(φ) Define the map ν by ν(ker(φ) + r ) = φ(r ).This is irrespective of the choice of coset representative. Since if ker(φ) + r = ker(φ) + s, then r − s ∈ ker(φ) so that φ(r ) = φ(s). Now check that ν is a ring homomorphism. We note that for any s ∈ S, we have that s = φ(r ) for some r ∈ R. Thus, ν(ker(φ) + r ) = φ(r ) = s and so ν is onto. Also, suppose ν(ker(φ) + r ) = ν(ker(φ) + s), then φ(r ) = φ(s) meaning r − s ∈ ker(φ) and ker(φ) + r = ker(φ) + s as desired. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 57 / 98 Examples 1 Considering the residue map Z → Z4 , the First isomorphism theorem gives the isomorphism B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 58 / 98 Examples 1 Considering the residue map Z → Z4 , the First isomorphism theorem gives the isomorphismZ4 ∼ = Z/h4i. 2 Consider the surjective map φ : Z × Z → Z defined by (a, b) 7→ b. The First isomorphism theorem gives the isomorphism B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 58 / 98 Examples 1 Considering the residue map Z → Z4 , the First isomorphism theorem gives the isomorphismZ4 ∼ = Z/h4i. 2 Consider the surjective map φ : Z × Z → Z defined by (a, b) 7→ b. The First isomorphism theorem gives the isomorphism(Z × Z)/hZ × {0}i ∼ = Z. 3 Consider the evaluation φ : R[x] → C defined by φ(f (x)) = f (i). Then by the First isomorphism theorem, B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 58 / 98 Examples 1 Considering the residue map Z → Z4 , the First isomorphism theorem gives the isomorphismZ4 ∼ = Z/h4i. 2 Consider the surjective map φ : Z × Z → Z defined by (a, b) 7→ b. The First isomorphism theorem gives the isomorphism(Z × Z)/hZ × {0}i ∼ = Z. 3 Consider the evaluation φ : R[x] → C defined by φ(f (x)) = f (i). Then by the First isomorphism theorem,R[x]/hx 2 + 1i ∼= C. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 58 / 98 Examples 1 Considering the residue map Z → Z4 , the First isomorphism theorem gives the isomorphismZ4 ∼ = Z/h4i. 2 Consider the surjective map φ : Z × Z → Z defined by (a, b) 7→ b. The First isomorphism theorem gives the isomorphism(Z × Z)/hZ × {0}i ∼ = Z. 3 Consider the evaluation φ : R[x] → C defined by φ(f (x)) = f (i). Then by the First isomorphism theorem,R[x]/hx 2 + 1i ∼= C. Try √ √ 1 Let φ : Z[ 5] → Z3 defined by φ(a + b 5) = [a − b]3. Determine if φ is a ring homomorphis. If yes, which rings are isomorphic under the first isomorphism theorem. 2 Show that Z/12Z is isomorphic to (Z/3Z) × (Z/4Z). B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 58 / 98 Theorem (Second Isomorphism Theorem) Let A be an ideal and B a subring of a ring R. Then, A + B = {a + b : a ∈ A, b ∈ B} is a subring of R and A ∩ B is an ideal of B. Furthermore, the rings (A + B)/A and B/A ∩ B are isomorphic. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 59 / 98 Theorem (Second Isomorphism Theorem) Let A be an ideal and B a subring of a ring R. Then, A + B = {a + b : a ∈ A, b ∈ B} is a subring of R and A ∩ B is an ideal of B. Furthermore, the rings (A + B)/A and B/A ∩ B are isomorphic. Diagramatic illustration RO A< + Bc Ac ; B A∩B B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 59 / 98 Theorem (Third Isomorphism Theorem) Let I , K be ideals in a ring R such that I ⊆ K. Then I is an ideal in R/I ∼ K , K /I is an ideal in R/I and = R/K. K /I B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 60 / 98 Theorem (Third Isomorphism Theorem) Let I , K be ideals in a ring R such that I ⊆ K. Then I is an ideal in R/I ∼ K , K /I is an ideal in R/I and = R/K. K /I Illustrations 1 In Z, considering the ideal 4Z and subring 6Z, by the second isomorphism theorem, we have that 2Z/4Z ∼ = 6Z/12Z. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 60 / 98 Theorem (Third Isomorphism Theorem) Let I , K be ideals in a ring R such that I ⊆ K. Then I is an ideal in R/I ∼ K , K /I is an ideal in R/I and = R/K. K /I Illustrations 1 In Z, considering the ideal 4Z and subring 6Z, by the second isomorphism theorem, we have that 2Z/4Z ∼ = 6Z/12Z. 2 Consider the ideals of 12Z and 2Z in Z. Which quotients are isomorphic? B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 60 / 98 Other properties of ideals Definition (Maximal Ideals) A maximal ideal of a ring R is an ideal M different from R such that there is no proper ideal N of R properly containing M. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 61 / 98 Other properties of ideals Definition (Maximal Ideals) A maximal ideal of a ring R is an ideal M different from R such that there is no proper ideal N of R properly containing M. Examples 1 The ideals pZ are maximal in Z. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 61 / 98 Other properties of ideals Definition (Maximal Ideals) A maximal ideal of a ring R is an ideal M different from R such that there is no proper ideal N of R properly containing M. Examples 1 The ideals pZ are maximal in Z. Proposition Let R be a ring with identity. Then every proper ideal is contained in a maximal ideal. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 61 / 98 Other properties of ideals Definition (Maximal Ideals) A maximal ideal of a ring R is an ideal M different from R such that there is no proper ideal N of R properly containing M. Examples 1 The ideals pZ are maximal in Z. Proposition Let R be a ring with identity. Then every proper ideal is contained in a maximal ideal. Theorem Let R be a commutative ring with identity. Then M is a maximal ideal if and only if R/M is a field. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 61 / 98 Definition (Prime Ideals) An ideal N 6= R in a commutative ring R is a prime ideal if ab ∈ N implies either a ∈ N or b ∈ N for a, b ∈ R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 62 / 98 Definition (Prime Ideals) An ideal N 6= R in a commutative ring R is a prime ideal if ab ∈ N implies either a ∈ N or b ∈ N for a, b ∈ R. Examples 1 The prime ideals in Z are precisely the ideals pZ for p prime. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 62 / 98 Definition (Prime Ideals) An ideal N 6= R in a commutative ring R is a prime ideal if ab ∈ N implies either a ∈ N or b ∈ N for a, b ∈ R. Examples 1 The prime ideals in Z are precisely the ideals pZ for p prime. 2 The ideal hxi in Z[x] is prime. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 62 / 98 Definition (Prime Ideals) An ideal N 6= R in a commutative ring R is a prime ideal if ab ∈ N implies either a ∈ N or b ∈ N for a, b ∈ R. Examples 1 The prime ideals in Z are precisely the ideals pZ for p prime. 2 The ideal hxi in Z[x] is prime. 3 The ideal h5i in Z[x] is prime. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 62 / 98 Definition (Prime Ideals) An ideal N 6= R in a commutative ring R is a prime ideal if ab ∈ N implies either a ∈ N or b ∈ N for a, b ∈ R. Examples 1 The prime ideals in Z are precisely the ideals pZ for p prime. 2 The ideal hxi in Z[x] is prime. 3 The ideal h5i in Z[x] is prime. Theorem Let R be a commutative ring with unity and let N 6= R be an ideal in R. Then R/N is an integral domain if and only if N is a prime ideal in R. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 62 / 98 Definition (Prime Ideals) An ideal N 6= R in a commutative ring R is a prime ideal if ab ∈ N implies either a ∈ N or b ∈ N for a, b ∈ R. Examples 1 The prime ideals in Z are precisely the ideals pZ for p prime. 2 The ideal hxi in Z[x] is prime. 3 The ideal h5i in Z[x] is prime. Theorem Let R be a commutative ring with unity and let N 6= R be an ideal in R. Then R/N is an integral domain if and only if N is a prime ideal in R. Remark Every maximal ideal is prime in a commutative ring. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 62 / 98 Polynomial rings Definition (Polynomial over a Ring) Let R be a ring. A polynomial over R in the indeterminate x is a formal sum ∞ X f (x) = ai x i = a0 + a1 x + a2 x 2 + · · · an x n + · · · i=0 where ai ∈ R and ai = 0 for all but finite number of values of i. The ai ’s are called the coefficients of f (x) and the largest i for which ai 6= 0 and aj = 0 for all j > i is called the degree of f (x) and corresponding coefficient ai is called the leading coefficient. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 63 / 98 Equality Two polynomials are equal by definition if they have the same degree and all corresponding coefficients are equal. Remark A polynomial function is f (x) from R to R is a defined by substituting elements of R in place of x. For example if f (x) = ∞ P i P∞ i i=0 ai x , then f (c) = i=0 ai c. We call f (c), the evaluation of f (x) at c. Two polynomial functions are equal if they yield the same value when evaluated at all elements of their domain. For example f (x) = x 5 − 2x + 1 and g (x) = 4x + 1 are equal as polynomial functions defined on Z5. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 64 / 98 Addition and Product of Polynomials P∞ P∞ If f (x) = i=0 ai x i and g (x) = f (x) = i=0 bi x i then ∞ X f (x) + g (x) = ci x i , where cn = an + bn. i=0 ∞ X n X i f (x)g (x) = di x where dn = ai bn−i i=0 i=0 We denote the set of polynomials in x over a ring R by R[x]. B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 65 / 98 Examples 1 In Z5 [x], 3x 4 (2x 2 + 4x + 3) = B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 66 / 98 Examples 1 In Z5 [x], 3x 4 (2x 2 + 4x + 3) = x 6 + 2x 5 + 4x 4 2 In Z2 [x], (x + 1)(x + 1) = B. O. Bainson (PhD) (Kwame Nkrumah University ABSTRACT of Science ALGEBRA and Technology) II Introduction to Rings July 11, 2023 66 / 98 Examples 1 In Z5 [x], 3x 4 (2x 2 + 4x + 3) = x 6 + 2x 5 + 4x 4 2 In Z2 [x], (x + 1)(x + 1) = x 2 + (1 + 1)x + 1 = x 2 + 1. 3 In Z2 [x], (x
