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.
Terminology 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 End R ( M ) = Hom R ( M , M ) {\displaystyle \operatorname {End} _{R}(M)=\operatorname {Hom} _{R}(M,M)} 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 category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.
Examples The zero map M → N that maps every element to zero. A linear transformation between vector spaces.
Hom Z ( Z / n , Z / m ) = Z / gcd ( n , m ) {\displaystyle \operatorname {Hom} _{\mathbb {Z} }(\mathbb {Z} /n,\mathbb {Z} /m)=\mathbb {Z} /\operatorname {gcd} (n,m)} . For a commutative ring R and ideals I, J, there is the canonical identification
Hom R ( R / I , R / J ) = { r ∈ R | r I ⊂ J } / J {\displaystyle \operatorname {Hom} _{R}(R/I,R/J)=\{r\in R|rI\subset J\}/J}
given by f ↦ f ( 1 ) {\displaystyle f\mapsto f(1)} . In particular, Hom R ( R / I , R ) {\displaystyle \operatorname {Hom} _{R}(R/I,R)} is the annihilator of I. Given a ring R and an element r, let l r : R → R {\displaystyle l_{r}:R\to R} denote the left multiplication by r. Then for any s, t in R,
l r ( s t ) = r s t = l r ( s ) t {\displaystyle l_{r}(st)=rst=l_{r}(s)t} . That is, l r {\displaystyle l_{r}} is right R-linear. For any ring R,
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