a ring

A ring is a set R and two binary operations, written + and ×, on R which satisfies the following conditions:

(R1) < R, + > is an abelian group with identity 0

(R2) × is associative.

(R3) × is distributive over +:

ie. a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a) for all a, b, c ∈ R.

(R4) there exists an element 1 ∈ R, different from 0, that is an identity for ×.

abelian

commutative, order doesn’t matter

subring

Let R be a ring and S ⊆ R. Then S is a subring of R if it is

a ring in its own right with respect to the same addition and multiplication as in R and S contains 1R

subring test

Let R be a ring and S ⊆ R. Then S is a subring of R, iff:

(i) 1R ∈ S;

(ii) r + s, r × s ∈ S, for all r, s ∈ S;

(iii) −r ∈ S for all r ∈ S

polynomial ring

Let R be a ring. The ring of polynomials R[X] in the indeterminate X is

defined as follows:

Elements: Formal linear combinations of the form

SUM [i≥0] aiX^i

where the coefficients ai ∈ R for i = 0, 1, … and only finitely many of these coefficients are non-zero. When writing specific polynomials we normally don’t include the terms with zero coefficients.

Equality:

SUM [i≥0] aiX^i = SUM [i≥0] biX^i ⇔ ai = bi for all i ≥ 0.

Addition:

SUM [i≥0] aiX^i + SUM [i≥0] biX^i =SUM [i≥0] (ai+bi) X^i

Multiplication:

[SUM [i≥0] aiX^i ] [SUM [i≥0] biX^i] =

SUM [k≥0] {SUM [i+j=k] aibj} X^k

The zero element is SUM [i≥0] 0X^i= 0

The one is 1X^0 +SUM [i≥0] 0X^i = 1

degree of polynomial

deg(f) = the largest i such that ai =! 0 and we let deg(f) = −∞ if f = 0.

cartesian product

If R1 and R2 are rings then the Cartesian product R1×R2

with operations + and × defined by

(r1, r2) + (s1, s2) = (r1 + s1, r2 + s2) and

(r1, r2) × (s1, s2) = (r1 × s1, r2 × s2) is a ring. The element (0, 0) is the zero of this ring and the element (1, 1) is the one of R1 × R2.

n · a

n · a = a + a + … + a (n times)

0 · a = 0 and setting n · a = (−n) · (−a) if n < 0. From the definition we get: (n + m) · a = n · a + m · a n · (a + b) = n · a + n · b n · (m · a) = (nm) · a where n, m ∈ Z and a, b ∈ R.

characteristic

The characteristic, char(R), of a ring R is the least positive integer n such that 1 + · · · + 1 = 0 (n 1s), that is, such that n · 1 = 0.

If there is no such n, then the characteristic of R is defined to be 0.

For example, Z has characteristic 0, as have the rings Q and C. On the otherhand char(Zn) = n

zero divisor

A non-zero element r ∈ R is a zero-divisor if there is a

non-zero element s ∈ R with rs = 0 or sr = 0.

domain

The ring R is a domain if, for all r, s ∈ R

rs = 0 ⇒ r = 0 or s = 0.

So a domain is a ring with no zero-divisors

A commutative domain is called an integral domain. (Recall that a ring R is said to be commutative if the multiplication is commutative: rs = sr for all r, s ∈ R.)

division ring

A division ring is a ring in which every non-zero element has a right inverse and a left inverse, ie:

for every r ∈ R there is s ∈ R such that rs = 1 (r is right invertible)

and there is t ∈ R such that tr = 1 (r is left invertible).

In this case (see below) s = t and we write r^-1 for this inverse of r and say just that r is invertible or that r is a unit.

field

A field is a commutative division ring

nilpotent

An element r of a ring R is nilpotent if there is some integer n >= 1 with r^n = 0

The least such n is the index of nilpotence of r

idempotent

An element r ∈ R is idempotent if r^2 = r. For example 0 and 1 are idempotent in any ring