Abstract Algebra Ring Theory Flashcards

Questions and Answers

What is a ring?

A set with two binary operations + and × (correct)

A nonempty set that is not a group

A subset of real numbers

A set with only one operation

What defines a ring with unity?

There exists an element 1_R in R such that for all a in R, 1_R * a = a and a * 1_R = a, with 1_R not equal to 0_R.

In a commutative ring, ab = ba for every a, b in R.

True

What is a zero-divisor?

An element a in R such that there exists a non-zero b in R for which ab = 0 or ba = 0. Signup and view all the answers

What does nilpotent mean in the context of rings?

An element a in R such that for some positive integer N, a^N = 0. Signup and view all the answers

Define a unit in a ring.

An element a in R such that there exists an element b in R with ab = 1 and ba = 1. Signup and view all the answers

What is a direct sum of rings?

A ring formed by the elements of the form (r₁, r₂) where r₁ is from R₁ and r₂ is from R₂. Signup and view all the answers

What is an integral domain?

A commutative ring with unity where the only zero-divisor is 0 itself. Signup and view all the answers

What characterizes a division ring?

All nonzero elements form a group under multiplication, and the ring has a unity. Signup and view all the answers

What is a field?

A commutative division ring where every nonzero element has an inverse. Signup and view all the answers

Define a subring.

A nonempty subset S of ring R that forms a ring under the operations of R. Signup and view all the answers

What is an ideal?

A nonempty subset I of R that is a subring and satisfies the left and right ideal conditions with every element of R. Signup and view all the answers

What is a quotient ring?

The set R/I, where I is an ideal of R, consisting of cosets of I in R. Signup and view all the answers

What is a principal ideal?

An ideal generated by a single element a in R by multiplying every element of R by a. Signup and view all the answers

What characterizes a prime ideal?

If ab is in I, then either a is in I or b is in I for a, b in R. Signup and view all the answers

What is a maximal ideal?

An ideal I that is not equal to R and is the largest proper ideal. Signup and view all the answers

What is the definition of a ring of polynomials?

R[x] = {a₀ + a₁x + ... + (a_n)xⁿ | a₀,...,a_n ∈ R, n ≥ 0}. Signup and view all the answers

Define a ring homomorphism.

A map phi from R₁ to R₂ that preserves addition and multiplication. Signup and view all the answers

What is the kernel of a ring homomorphism?

The set of elements in R that map to the additive identity in R₂. Signup and view all the answers

What is a prime subfield?

A subfield that is isomorphic to either the rational numbers Q or the finite field (Zp,⊕,⊗). Signup and view all the answers

What does characteristic zero mean?

The field has a subfield isomorphic to Q, meaning 1_F has infinite order. Signup and view all the answers

What is characteristic p?

The field has a subfield isomorphic to Zp, where p is prime. Signup and view all the answers

What are coefficients in polynomials?

The numbers a₀, a₁,..., a_n that multiply the variable x raised to different powers. Signup and view all the answers

What is a root of a polynomial?

A value a in F such that f(a) = 0 for a polynomial f(x). Signup and view all the answers

What does irreducible mean in the context of polynomials?

A nonconstant polynomial that cannot be factored into the product of two polynomials of lesser degree. Signup and view all the answers

Study Notes

Ring Theory Concepts

Ring: A nonempty set R with two operations (+, ×) where (R,+) is an abelian group, multiplication is associative, and distributes over addition. Example: Rational numbers (Q).
Ring With Unity: A ring containing a multiplicative identity element 1_R, satisfying (1_R)a = a(1_R) = a for all a in R, and 1_R ≠ 0_R. Example: Integers modulo n (Z_n) for n ≥ 1.
Commutative Ring: A ring where multiplication is commutative (ab = ba for all a, b in R). Example: Even integers under addition and multiplication.
Zero-Divisor: An element a in a ring R is a zero-divisor if there exists a nonzero b in R such that ab = 0 or ba = 0. Example: In Z₆, both 2 and 3 are zero-divisors since 2 × 3 = 0.
Nilpotent: An element a in a ring R is nilpotent if a^n = 0 for some positive integer n. Example: In Z₂₇, the element 6 is nilpotent because 6³ = 0.
Unit: An element a in a ring R is a unit if there exists an element b in R such that ab = 1_R. Example: In Z₇, the element 4 is a unit since 4 × 2 = 1.

Advanced Ring Structures

Direct Sum: The direct sum of rings R₁ and R₂, denoted R₁ ⊕ R₂, consists of elements (r₁, r₂) with operations defined component-wise. Example: Z ⊕ Z.
Integral Domain: A commutative ring with unity where the only zero-divisor is 0. Examples include the set of real numbers, rationals, and integers.
Division Ring: A ring in which every nonzero element has a multiplicative inverse and includes a unity. All nonzero elements form a multiplicative group.
Field: A commutative division ring, where every nonzero element has an inverse. Example: Finite fields like Z_p where p is prime.

Ideal Theory

Subring: A nonempty subset S of a ring R is a subring if S itself is a ring under the operations of R. Example: Integers (Z) as a subring of real numbers (R).
Ideal: A nonempty subset I of a ring R is an ideal if I is a subring and satisfies the condition that ra ∈ I and ar ∈ I for all a in I and r in R. Example: The ideal of even integers in Z.
Quotient Ring: Defined as R/I, where I is an ideal, consisting of cosets of I in R with corresponding operations. Example: Q[x]/(x²) relates to polynomial rings.
Principal Ideal: An ideal generated by a single element a in R, denoted as aR.
Prime Ideal: An ideal I such that if ab ∈ I, then either a ∈ I or b ∈ I. Example: In Z, the ideal (n) for prime n.
Maximal Ideal: An ideal I that is the largest proper ideal of R. If I ⊆ J ⊆ R, then J is either I or R. Example: In Z, (p) for prime p is maximal.

Polynomial Rings and Fields

Ring of Polynomials: Denoted R[x], consists of polynomials with coefficients from R. If R has unity, so does R[x]. Example: Polynomials with integer coefficients.
Ring Homomorphism: A map between rings preserving the operations: φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b). Example: Mapping integers to integers modulo n.
Kernel: The kernel of a ring homomorphism φ is the set of elements that map to the zero element in the codomain. It is also an ideal.

Field Characteristics

Prime Subfield: A subfield isomorphic to either the rationals Q or the finite field Z_p for prime p.
Characteristic Zero: A field F is of characteristic zero if it contains a subfield isomorphic to Q. Examples include Q, R, and C.
Characteristic p: A field F has characteristic p if it contains a subfield isomorphic to Z_p. Every finite field has some prime characteristic.

Other Key Concepts

Coefficients: In a polynomial, the degree is n if the leading coefficient (a_n) is non-zero. The degree of the zero polynomial is undefined.
Root: An element a is a root of the polynomial f(x) if f(a) = 0. Example: In Z₂, 1 is a root of f(x) = x² + 1.
Irreducible: A nonconstant polynomial that cannot be factored into the product of two lower-degree polynomials. Example: x² + 1 is irreducible in R[x].

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Test your knowledge of ring theory concepts in abstract algebra with these informative flashcards. Each card presents fundamental definitions and examples of rings, ring with unity, and commutative rings. Perfect for students looking to reinforce their understanding of the topic.