Answer to Question #12708 in Abstract Algebra for Hym@n B@ss

Question

Prove that all ideals in field are trivial.

Answer 1

Proof.
Let F be a field and J be a non-zero ideal of F, i.e. J is not
{0}.
We should prove that J=F.

Let u be a non-zero element of J, and

w = u^{-1}
be its inverse, so
wu = 1.

Then
1 = wu
belongs to J,
wehnce for any a from F we have that
a * 1 = a belogs to
J.

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