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 ${\displaystyle f:M\to N}$ is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,

${\displaystyle f(x+y)=f(x)+f(y),}$

${\displaystyle f(rx)=rf(x).}$

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

${\displaystyle f(xr)=f(x)r.}$

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 ${\displaystyle \operatorname {Hom} _{R}(M,N)}$. 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.