Abstract Algebra: Rings & Ideals Quiz

Rings

• A ring (with unity) consists of a set with operations * and + where (R, +) is an abelian group and includes multiplicative identity 1.
• Operations follow distributive properties: a * (b + c) = ab + ac and (b + c)a = ba + c*a.
• A ring is commutative if a * b = b * a for all a, b in R.

Types of Rings

• Examples of commutative rings include:
  • Z (integers)
  • Z ln Z
  • IR (real numbers)
  • R[x] (polynomials with real coefficients)
  • Q (rational numbers)
• A zero divisor is a non-zero element a in R such that there's a non-zero b in R with ab = 0.
• An integral domain is a commutative ring without zero divisors.

Fields

• A field is a commutative ring where every non-zero element has a multiplicative inverse.
• Examples of fields include:
  • Q (rationals)
  • IR (reals)
  • Z/pZ for prime p.

Subrings and Ideals

• A subring S of R satisfies:
  • Contains unity (1 ∈ S).
  • (S, +) is a subgroup of (R, +).
  • For a, b in S, their product ab is also in S.
• An ideal I of R is defined by:
  • (I, +) being a subgroup.
  • For x ∈ I, r ∈ R, both rx and xr belong to I.

Prime Ideals

• Ideal I in a commutative ring R is prime if ab ∈ I implies a ∈ I or b ∈ I.
• Examples of prime ideals:
  • pZ in Z for a prime p.
  • Irreducible polynomials in F[x].

Homomorphisms

• A (unital) ring homomorphism f: R → S includes:
  • f(1) = 1.
  • f(a + b) = f(a) + f(b).
  • f(ab) = f(a)f(b).
• Isomorphism exists if the homomorphism is bijective.
• The kernel of f, Ker(f), consists of elements in R mapped to the zero element in S.

Additional Concepts

• The characteristic of R is the smallest n such that adding 1 n times equals 0.
• For any ring, a unique homomorphism exists from Z to R with Kernel char(R)Z.
• Every ideal in Z and F[x] is principal (generated by a single element).
• Fundamental Homomorphism Theorem states R/Ker(f) ≅ S for a surjective homomorphism f.

Groups

• A group G has an operation * satisfying:
  • Associativity: a * (b * c) = (a * b) * c.
  • There exists an identity element e such that e * g = g for all g in G.
  • Each g has an inverse g^(-1) such that g * g^(-1) = e.
• A group is abelian if a * b = b * a, and examples include Z and S_n (symmetric groups).

Subgroups and Orders

• A subgroup H of G must satisfy:
  • H is non-empty.
  • For a, b in H, the product ab is in H.
  • If a is in H, then the inverse a^(-1) is also in H.
• The order of g in G is the smallest positive integer n such that g^n equals the identity.

Isomorphisms and Kernels

• A map f: G → H is a homomorphism if f(ab) = f(a)f(b).
• The kernel of a homomorphism f: G → H is Ker(f) = { g in G | f(g) = e }.