A module homomorphism is a map between modules over a ring
which preserves both the addition and the multiplication by
scalars. In symbols this means that
and
Note that if the ring is replaced by a field
, these conditions yield exactly the definition of
as a linear map between abstract vector spaces over
.
For all modules
over a commutative ring
, and all
, the multiplication by
determines a module homomorphism
, defined by
for all
.