IDA: Interactive Document on Algebra

Section 7.5
Ideals

Ideals appear in the study of ring homomorphisms. They are very useful in the study of polynomial equations, and in the construction of rings by means of residue classes, in much the same way we have seen them come about in modular and polynomial arithmetic.

As before, we only consider commutaive rings. So, let R be a commutative ring.

Definition

A nonempty subset I of R is an ideal of R if, for all a, b I and all r R,

a + b

I, ra

Each ideal contains 0. The subsets {0} and R of R are both ideals of R.

If a subset V of R is contained in an ideal I, then every combination

r₁v₁ + r₂ v₂ + ··· + r_nv_n (with r₁, ..., r_n

R and v₁, ..., v_n

also belongs to I. In fact, all these combinations form an ideal themselves.

Theorem

Let V be a nonempty subset of R. The subset of R consisting of all combinations of the form

r₁v₁ + r₂ v₂ + ··· + r_n v_n,

with r₁, ..., r_n R and v₁, ..., v_n V, is an ideal of R.

It is called the ideal generated by V. Notation (V) or VR.