Abstract Algebra Ring Theory Flashcards

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].