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Concept
Module homomorphism
Summary
In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,
In other words, f is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If M, N are right R-modules, then the second condition is replaced with
The 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 . It is an abelian group (under pointwise addition) but is not necessarily a module unless R is commutative.
The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorphism. Thus, all the (say left) modules together with all the module homomorphisms between them form the .
A module homomorphism is called a module isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups.
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 of a module M. It is not only an abelian group but is also a ring with multiplication given by function composition, called the endomorphism ring of M. The group of units of this ring is the automorphism group of M.
Schur's lemma says that a homomorphism between simple modules (modules with no non-trivial submodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a division ring.
In the language of the , an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.
Official source
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