MAT2006: Groups, Rings & Fields The ring axioms

A ring R is a set R together with operations of addition and multiplication on R, that is, given a, b ∈ R there are elements a + b, ab ∈ R, satisfying the following conditions:

A1 a + b = b + a for all a, b ∈ R. (The commutative law for addition.)

A2 (a + b) + c = a + (b + c) for all a, b, c ∈ R. (The associative law for addition.)

A3 There is an element 0 ∈ R such that a + 0 = a for all a ∈ R. (There is an additive identity.)

A4 For each a ∈ R there is an element −a ∈ R with a + (−a) = 0. (There exist additive inverses.)

M1 (ab)c = a(bc) for all a, b, c ∈ R. (The associative law for multiplication.)

M2 There is an element 1 ∈ R such that a1 = a = 1a for all a ∈ R. (There is a multiplicative identity.)

D a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c ∈ R. (The distributive law.)

Note that axioms A1–4 say that R is an Abelian group under the operation of addition.

We do not assume that ab = ba, but a ring for which this is always true is called a commutative ring.

A field is a commutative ring where the nonzero elements form a group under multiplication, that is 1 ≠ 0 and for each a ≠ 0 there is b ∈ R with ab = 1.

RJC 22/11/2000