Editing Module homomorphism (section)

{{short description|Linear map over a ring}}In [[Abstract algebra|algebra]], a '''module homomorphism''' is a [[function (mathematics)|function]] between [[module (mathematics)|module]]s that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a [[Ring (mathematics)|ring]] ''R'', then a function <math>f: M \to N</math> is called an ''R''-''module homomorphism'' or an ''R''-''linear map'' if for any ''x'', ''y'' in ''M'' and ''r'' in ''R'',

:<math>f(x + y) = f(x) + f(y),</math>
:<math>f(rx) = rf(x).</math>
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
:<math>f(xr) = f(x)r.</math>

The [[preimage]] of the zero element under ''f'' is called the [[kernel (algebra)|kernel]] of ''f''. The [[Set (mathematics)|set]] of all module homomorphisms from ''M'' to ''N'' is denoted by <math>\operatorname{Hom}_R(M, N)</math>. It is an [[abelian group]] (under pointwise addition) but is not necessarily a module unless ''R'' is [[Commutative ring|commutative]].

The [[Function composition|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]].