# Endomorphism ring of an abelian group

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## Definition

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.

## Facts

- 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.