Module homomorphism
In algebra, a module homomorphism is a function between modules that preserves module structures. Explicitly, if M and N are left modules over a ring R, then a function is called a module homomorphism or an R-linear map if for any x, y in M and r in R,
If M, N are right modules, then the second condition is replaced with
The pre-image of the zero element under f is called the kernel of f. The set of all module homomorphisms from M to N is denoted by HomR(M, N). It is an abelian group but is not necessarily a module unless R is commutative.
The composition of module homorphisms is again a module homomorphism. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules.
A module homomorphism is called an isomorphism if it admits the inverse homomorphism. A module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups. In other words, an inverse function of a module homomorphism, when it exists, is necessary a homomorphism.
The isomorphism theorems hold for module homomorphisms.
A module homomorphism from a module M to itself is called an endomorphism and an isomorphism from M to itself an automorphism. One writes for the set of all endomorphisms between a module M. It is not only an abelian group but is also a ring with multiplication given by function composition; it is called the endomorphism ring of M.
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