Let R be a ring. Just like with subrings and submonoids, we can also describe generation of ideals by means of intersections.

Theorem

If C is a collection of ideals of R, then

I C I

is also an ideal of R.

Now, if V is a subset of R, the ideal generated by R is the same as the intersection of all ideals containing V. Note that this collection is nonempty, as R itself is an ideal.

The following is a characterization of this special ideal.

Proposition

Suppose I is an ideal of R. The following are equivalent.

  1. I = R.
  2. 1 I.
  3. I contains an invertible element.
  4. There are v1, ..., vn I and r1, ..., rn R such that 1 = r1, ..., rn I such that

    1 = r1v1 + ··· + rnvn.