Hom, written 
for left unital
and 
over a unit ring 
, is the set of all module
homomorphisms from 
to 
. It is an Abelian
group under pointwise addition,
 |
(1)
|
If 
is commutative,
it is also an 
-module under the operation
 |
(2)
|
for all 
(Knapp 2016, p. 554). For a noncommutative ring, this
operation need not preserve module homomorphisms.
Indeed, 
-linearity of 
would require 
for all 
and 
,
which need not hold.
denotes the covariant
functor from the category of left unital
R-modules to the category of Abelian
groups. It maps every module 
to 
and every module homomorphism
 |
(3)
|
to the group homomorphism
 |
(4)
|
defined, for all 
, by
 |
(5)
|
The contravariant functor 
also takes values in Abelian
groups. It maps 
to 
and the module
homomorphism 
to the group homomorphism
 |
(6)
|
defined, for all 
, by
 |
(7)
|
When 
is commutative,
both functors can instead be viewed as taking values
in 
-modules with
the operation defined above.
