Answer to Question #12457 in Abstract Algebra for john.george.milnor

Question

Show that invertible elements form a group in any associative ring.

Answer 1

Let R be any associative ring with identity.
Let U be group of invertible
elements of ring R.
As by the ring definition

a(bc)=(ab)c
a*1=1*a=a
where any a,b,c, from U.
So we need to
concentrate on third axiom - any element have to have its inverse.
From
definition of invertible element: a is invertible iff there is some b that
ab=ba=1.
This also mean that U is group under maltiplication.