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.