Ideal (ring Theory) - Ideal Generated By A Set

Ideal Generated By A Set

Let R be a (possibly not unital) ring. Any intersection of any nonempty family of left ideals of R is again a left ideal of R. If X is any subset of R, then the intersection of all left ideals of R containing X is a left ideal I of R containing X, and is clearly the smallest left ideal to do so. This ideal I is said to be the left ideal generated by X. Similar definitions can be created by using right ideals or two-sided ideals in place of left ideals.

If R has unity, then the left, right, or two-sided ideal of R generated by a subset X of R can be expressed internally as we will now describe. The following set is a left ideal:

Each element described would have to be in every left ideal containing X, so this left ideal is in fact the left ideal generated by X. The right ideal and ideal generated by X can also be expressed in the same way:

The former is the right ideal generated by X, and the latter is the ideal generated by X.

By convention, 0 is viewed as the sum of zero such terms, agreeing with the fact that the ideal of R generated by ∅ is {0} by the previous definition.

If a left ideal I of R has a finite subset F such that I is the left ideal generated by F, then the left ideal I is said to be finitely generated. Similar terms are also applied to right ideals and two-sided ideals generated by finite subsets.

In the special case where the set X is just a singleton {a} for some a in R, then the above definitions turn into the following:

These ideals are known as the left/right/two-sided principal ideals generated by a. It is also very common to denote the two-sided ideal generated by a as (a).

If R does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in X with things in R, we must allow the addition of n-fold sums of the form x+x+...+x, and n-fold sums of the form (−x)+(−x)+...+(−x) for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous.