Acronyms that contain the term module homomorphism 

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 f : M → N {\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, f ( x + y ) = f ( x ) + f ( y ) , {\displaystyle f(x+y)=f(x)+f(y),} f ( r x ) = r f ( x ) . {\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 f ( x r ) = f ( x ) r . {\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 Hom R ⁡ ( M , N ) {\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.

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