Ideals. In accordance with previous use of the term, a subring of a ring R is a set S of elements of R such that, with respect to the addition and multiplication defined in R, S is itself a ring. Previously mentioned examples of subrings are the subring of all even integers in the ring of all integers, the subring of all integers in the field of rational numbers, and the subring of all finite subsets of J in the ring of all subsets of J.

Although subrings are sometimes of interest, it will be shown presently that subsystems of more restricted type are of even more importance. These are the ideals which we proceed to define.

A set S of one or more elements of a ring R is called an ideal in R if and only if it has the following properties:

(i) If a and b are elements of S, then a — b is an element of S.

(ii) If a is an element of S, then for every element r of R, ar and ra are elements of S.

We may first point out that an ideal S in R is necessarily a subring of R. By (i), if a is an element of S, then 0 = a — a is in S, and it then follows that — a = 0 — a is also in S.