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

Question #12457

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

Expert's answer

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.

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.

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