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

Question #12707

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

Expert's answer

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.

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.

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