Algebra Handout 2: Ideals and Quotients

Remark (Ideals versus subrings): It is worthwhile to compare these two notions; they are related, but with subtle and important differences. Both an ideal I and a subring S of a ring R are subsets of R which are subgroups under addition and are stable under multiplication. However, each has an additional property: for an ideal it is the absorption property (IR2). For instance, the integers Z are a subring of the rational numbers Q, but are clearly not an ideal, since 1 2 ·1 = 1 2 , which is not an integer. On the other hand a subring has a property that an ideal usually lacks, namely it must contain the unity 1 of R. For instance, the subset 2Z = {2n | n ∈ Z} is an ideal of Z but is not a subring.