Answer to Question #12707 in Abstract Algebra for Hym@n B@ss

Question

Prove that every ideal with invertible element inside is trivial ideal of a given ring.

Answer 1

Proof.
Let R be a ring and J be an ideal of R, so

aj belongs to
J

for all a from R and j from J.

Suppose u is an invertible
element of R belonging to I, so there exists
w = u^{-1}
such that

wu = 1.

But u belongs to J, whence
1 = wu belongs to
J.

Hence for any a from R we have that
a * 1 = a belogs to
J.

That is J=R, and so J is a trivial ideal of R.