opposite ring

Definition

For any ring , the opposite of , denote , is the ring with the elements of with the structure of an abelian group. The group multiplication is reversed, that is

where the left hand side is the multiplication in , and the right hand side is multiplication in .

Equivalence to endomorphism ring

Proof

For a ring define the map

where (that is, it is right multiplication by ).

Then we have

Thus, we have a homomorphism

Note if then

so and the homomorphism is injective.

To show surjectivity, let , we claim that because