Module homomorphism  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 formula_1 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 preimage 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 formula_5. 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 category of modules. 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 formula_6 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, 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 (a module with no non-trivial submodules) must be either zero or an isomorphism.