Endomorphism ring of an abelian group
Suppose is an abelian group. The endomorphism ring of , denoted is defined as follows:
- As a set, it is the set of all endomorphisms of .
- The addition is pointwise addition in the target group. In other words, for endomorphisms of , we define as the map . Thus, the additive identity is the zero map (the map sending everything to zero) and the negation is the pointwise negation.
- The multiplication is given by function composition. In other words, is the map sending to . The identity for multiplication is the identity map.
- For a non-abelian group, the pointwise group multiplication of two endomorphisms need not be an endomorphism.
- For functions that are not endomorphisms, only one of the distributivity laws holds in general.