Abstract Algebra Final

Well-Ordering Axiom

States that every nonempty subset of the natural numbers has a least element.

“b divides a”

For integers a and b, we say that b divides a if there exists an integer k such that a = bk.

Prime Number

It is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

a ≡ b (mod n)

For integers a, b, and positive integer n, it states that n divides the difference a − b.

Congruence class mod n

It is the set of all integers that are congruent to a modulo n; that is, it is the set: [a]_n = {x ∈ Z | x ≡ a (mod n)}.

Commutative Ring

It is a set R equipped with two binary operations, addition and multiplication, such that R is an abelian group under addition, multiplication is associative and commutative, and the distributive laws relate the two operations.

Ring with Identity

It is a ring R that contains a multiplicative identity element, denoted 1, such that 1r = r1 = r for every element r of R.

Integral Domain

It is a commutative ring with identity in which the product of two nonzero elements is never zero; equivalently, it is a ring with no zero divisors.

Field

It is a commutative ring with identity in which every nonzero element has a multiplicative inverse, meaning that there exists an element a^(-1) such that aa^(-1) = 1.

Zero Divisor

A zero divisor in a ring R, is a nonzero element a for which there exists a nonzero element b in R such that ab = 0

Congruence Class Modulo (an Ideal I in a ring R)

Given a ring R and an ideal I of R, the congruence class of an element a modulo I is the set a + I = { a + x | x ∈ I} which consists of all elements of R that differ from a by an element of the ideal I.

Principal Ideal

A principal ideal of a ring R is an ideal generated by a single element; specifically if there exists an element a ∈ R such that I = (a) = { ra | r ∈ R }.

Kernel of a Homomorphism

Let f : R→S be a homomorphism of rings. The kernel of f is k = {r ∈ R | f(r) = 0_S​}.

Prime Ideal

An ideal P of a commutative ring R is called a prime ideal if P ≠ R and whenever a product ab lies in P, then at least one of the elements a or b lies in P.