The concept of an “ideal” in a ring is analogous to the concept of a “normal subgroup” in a group. Rings modulo an ideal are constructed in the same canonical fashion as groups modulo a normal subgroup. The role of an ideal in a “homomorphism between rings” is similar to the role of a normal subgroup in a “homomorphism between groups.” Theorems proved in this chapter on the direct sum of ideals in a ring and on homomorphisms between rings are parallel to the corresponding theorems for groups proved in Chapters 5 and 8.

Ideals

Definition.A nonempty subset S of a ring R is called an ideal of R if

1. (i) a,b ∈ S implies a - b ∈ S.

2. (ii) a ∈ S and r ∈ R imply ar ∈ S and ra ∈ S.

Definition.A nonempty subset S of a ring R is called a right (left) ideal if

1. (i) a,b ∈ S implies a - b ∈ S.

2. (ii) ar ∈ S (ra ∈ S) for all a ∈ S and r ∈ R.

Clearly, a right ideal or a left ideal is a subring of R, and every ideal is both right and left, so an ideal is sometimes called a two-sided ideal. Trivially, in a commutative ring every right ideal or left ideal is two-sided.

In every ring R, (0) and R are ideals, called trivial ideals.

Examples of ideals

(a) In the ring of integers Z every subring is an ideal. To see this, let I be a subring of Z and a ∈ I, r ∈ Z.