Rings and Subrings

• A ring is an algebraic structure (R, +, .) where R is a nonempty set with two operations + and .

Ring Conditions

• (R, +) must be an abelian group
• (R, .) must be a semi-group
• Distributive laws must hold: a.(b + c) = a.b + a.c and (a + b).c = ac + bc for all a, b, c in R

Subring

• If S is a subset of ring R, then S is a subring of R if S is also a ring with the same operations as R

Subring Test

• A nonempty subset S of ring R is a subring if:
• a - b is in S for all a, b in S
• ab is in S for all a, b in S

Commutative Ring

• If multiplication is commutative in R (i.e., a.b = b.a for all a, b in R), then (R, +, .) is a commutative ring
• Examples: (2Z, +, .) and (nZ, +, .) where n > 1

Ring with Unity

• If R contains a multiplicative identity element (unity), denoted as 1 where 1 ≠ 0 (additive identity), implies (R, +, .) is a ring with unity.
• Example: Mn(R, +, .) is a ring with unity.

Commutative Ring with Unity

• If multiplication is commutative in R and R has a multiplicative identity 1, then R is a commutative ring with unity.

Unit in a Ring

• For a ring (R, +, .), element a ≠ 0 in R is a unit if there exists a non-zero element b in R such that ab = 1
• a is a unit in R if a has a multiplicative inverse in R.

Zero Divisors

• In ring (R, +, .), if a, b are in R, and a ≠ 0 and b ≠ 0, but ab = 0, then a and b are zero divisors
• Example: In the ring (M2(R, +, .)), certain matrices are zero divisors because their product is the zero matrix, even though neither matrix is zero.

Finite Commutative Ring with Unity

• In a finite commutative ring with unity, every non-zero element is either a unit or a zero divisor
• Cancellation laws with respect to multiplication hold in a ring if it has no zero divisors

Integral Domain

• Commutative ring with unity with no zero divisors
• Integral domain is a commutative ring with unity without zero divisors.
• Examples: (Z, +,.) and Z[i] = {a + ib : a, b ∈ Z}

Skew Field (Division Ring)

• Ring with unity where every non-zero element is a unit
• Field
• A ring (R,+,.) where (R*,.) is an abelian group is a field
• A field is a commutative division ring

Subfield

• If F is a field, a subset E of F is a subfield if E itself is a field with the same operations as F

Subfield Test

• A nonempty subset E of a field F is a subfield if:
• a - b is in E
• ab^(-1) is in E for all a, b in E

Results

• Every field is an integral domain
• Every finite integral domain is a field
• Every finite division ring is a field
• A non-commutative division ring is called strictly skew field
• Any strictly skew field is infinite
• (Zn,+n, xn) is a field iff n is a prime in Z

Cross Product of Rings

• Given two rings (R1, +1,x1) and (R2, +2, x2), the Cartesian product R1 × R2 is also a ring under specific operations
• Operations:
• (r1, s1) + (r2, s2) = (r1 +1 r2, s1 +2 s2)
• (r1, s1) × (r2, s2) = (r1 ×1 r2, s1 ×2 s2)

Results Regarding Cross Product

• The cross product of two integral domains does not necessarily result in an integral domain.
• Example: Z is an integral domain, but Z × Z is not an integral domain.
  • (1,0)(0,1) = (0,0)
• The cross product of two fields does not necessarily result in a field.
• Example: R is a field, though R × R is not.

Characteristic of a Ring

• The characteristic of a ring (R, + , .) is the smallest positive integer n such that na = 0 for every a in R.
• Denoted as char(R).
• If no such positive integer exists, then char(R) = 0.

Examples of Ring Characteristics

• char(Q) = char(R) = char(Z) = 0
• char(Zn) = n
• char(Zn × Zm) = lcm(char(Zn), char(Zm)) = lcm(n, m)
• char(Z × Zn) = 0
• char(Mn(F)) = char(F)
• char(Mn(Zp)) = p

Field of Quotients of an Integral Domain

• If D is an integral domain but not a field, it can be extended to a field F such that every element of F is a quotient of two elements of D
• Fis the smallest field containing D, and is the intersection of all fields containing D

Operations in Field of Quotients

• Given a/b and c/d in F (where a, b, c, d ∈ D and b, d ≠ 0):
• a/b + c/d = (ad + bc) / bd
• (a/b) * (c/d) = ac / bd

Examples of Field of Quotients

• The field of quotients of Z is Q
• The field of quotients of Z[i] is Q(i)
• The field of quotients of R[x] is R(x) (field of rational functions)

Ideals

• If R is a ring and I is a subring of R, then I is a left ideal of R if for every r in R and a in I, ra is in I, equivalent to rI ⊆ I
• Similarly, I is a right ideal of R if for every r in R and a in I, ar is in I, equivalent to Ir ⊆ I
• If I is both a left and right ideal of R, then I is simply called an ideal of R, so rI ⊆ I and Ir ⊆ I for all r in R

Examples of Ideals

• The ideals of the ring (Z, +, .) are in the form (nZ, +, .) for n ∈ Z
• The ideals of the ring (Zn, +n, ×n) are in the form (mZm, +n, ×n) for m ∈ Zn

Ideal Test

• A nonempty subset I of a ring R is an ideal of R if:
• a − b ∈ I and ab ∈ I for all a, b ∈ I
• ra, ar ∈ I for all a ∈ I, r ∈ R

Result Regarding Ideals in Fields

• A field F has only trivial (improper) ideals, which are F itself and {0}

Simple Ring

• A ring R with no proper non-trivial ideals

Examples of Simple Rings

• Every field is a simple ring
• Mn(F), where F is a field, is a simple ring

Ideals in Product Rings

• If I1 is an ideal of ring R1 and I2 is an ideal of ring R2, then I1 × I2 is an ideal of R1 × R2

Rings and Ideals with Unity

• If R is a ring with unity 1 ≠ 0 and I is an ideal of R such that 1 ∈ I, then I = R
• If R is a ring with unity 1 ≠ 0, and I is an ideal of R containing a unit u (u ∈ I), then I = R

Principal Ideal

• Within commutative ring R with unity, if a ∈ R, then the set is an ideal of R called a principal ideal generated by a
• Ideal I is called a principal ideal if it is generated by a single element of R
• The ideals of (Z, +, .) are all principal ideals, and (nZ, +, .) =

Results about Principal Ideals

• If F is a field, then all its ideals are principal ideals: {0} = and F =
• If F is a field, all ideals of F[x] are principal ideals
• The ideals in (Zn, +n, ×n) are principal ideals

Prime Ideal

• Given a commutative ring R with unity, ideal I is a prime ideal if it is a proper ideal of R such that for a, b ∈ R, a·b ∈ I implies either a ∈ I or b ∈ I

Examples of Prime Ideals

• In Z, the ideal pZ is a prime ideal where p is a prime number in Z
• is a prime ideal of Z
• is not a prime ideal of Z6 because 2, 3 ∈ Z6 and 2 × 3 = 0 ∈, but 2 and 3 are not in
• is a prime ideal of ring R if R has no zero divisors

Maximal Ideal

• A proper ideal I of commutative ring R (with unity) is a maximal ideal if no ideal exists that properly contains I and is properly contained in R
• I ⊂ A ⊂ R implies either A = R or A = I

Relation Between Maximal and Prime Ideals

• Every maximal ideal is a prime ideal; converse is generally not true
• In Z, is a prime ideal, but it is not maximal since
• The maximal ideals of Z are where p is a prime

Ring Homomorphism

• A mapping φ of ring R into ring R' is a homomorphism if it preserves addition and multiplication
• φ(a + b) = φ(a) + φ(b)
• φ(ab) = φ(a)φ(b)
• This must hold for all a , b in R

Kernel and Image of Ring Homomorphism

• Let φ : R1 → R2 be a ring homomorphism. Then the kernel of φ (ker φ) is {a ∈ R1 : φ(a) = 0 in R2}
• ker φ is a subring of R1
• The image of φ (Imp) is {b ∈ R2 : b = φ(a) for some a ∈ R1}
• Imp is a subring of R2