suppose we have a field F (which is, of course, also a ring). let I be an ideal of F that contains x ≠ 0.

since x ≠ 0, we have 1/x in F, and since I is an ideal 1 = (1/x)(x) is also in I.

thus for ANY a in F, a = (a)(1) is in I, so that I = F.

(in any ring with identity, if an ideal I contains 1, it is the entire ring).