the range of a homomorphism is a sunmodule
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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}f:M\to N is called a 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),}f(x+y)=f(x)+f(y),
{\displaystyle f(rx)=rf(x).}f(rx)=rf(x).
If M, N are right R-modules, then the second condition is replaced with
{\displaystyle f(xr)=f(x)r.}f(xr)=f(x)r.
The pre-image 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)}\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. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules.
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Terminology Edit
A module homomorphism is called an isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. 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 {\displaystyle \operatorname {End} _{R}(M)=\operatorname {Hom} _{R}(M,M)}{\displaystyle \operatorname {End} _{R}(M)=\operatorname {Hom} _{R}(M,M)} 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 having only two 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 Edit
The zero map M → N that maps every element to zero.
A linear transformation between vector spaces.
{\displaystyle \operatorname {Hom} _{\mathbb {Z} }(\mathbb {Z} /n,\mathbb {Z} /m)=\mathbb {Z} /\operatorname {gcd} (n,m)}\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
{\displaystyle \operatorname {Hom} _{R}(R/I,R/J)=\{r\in R|rI\subset J\}/J}{\displaystyle \operatorname {Hom} _{R}(R/I,R/J)=\{r\in R|rI\subset J\}/J}
given by {\displaystyle f\mapsto f(1)}f\mapsto f(1). In particular, {\displaystyle \operatorname {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 {\displaystyle l_{r}:R\to R}{\displaystyle l_{r}:R\to R} denote the left multiplication by r. Then for any s, t in R,
{\displaystyle l_{r}(st)=rst=l_{r}(s)t}{\displaystyle l_{r}(st)=rst=l_{r}(s)t}.
That is, {\displaystyle l_{r}}{\displaystyle l_{r}} is right R-linear.
For any ring R,
{\displaystyle \operatorname {End} _{R}(R)=R}\operatorname {End}_{R}(R)=R as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation {\displaystyle R{\overset {\sim }{\to }}\operatorname {End} _{R}(R),\,r\mapsto l_{r}}{\displaystyle R{\overset {\sim }{\to }}\operatorname {End} _{R}(R),\,r\mapsto l_{r}}.
{\displaystyle \operatorname {Hom} _{R}(R,M)=M}\operatorname {Hom}_{R}(R,M)=M through {\displaystyle f\mapsto f(1)}f\mapsto f(1) for any left module M.[1] (The module structure on Hom here comes from the right R-action on R; see #Module structures on Hom below.)
{\displaystyle \operatorname {Hom} _{R}(M,R)}\operatorname {Hom}_{R}(M,R) is called the dual module of M; it is a left (resp. right) module if M is a right (resp. left) module over R with the module structure coming from the R-action on R. It is denoted by {\displaystyle M^{*}}M^*.
Given a ring homomorphism R → S of commutative rings and an S-module M, an R-linear map θ: S → M is called a derivation if for any f, g in S, θ(f g) = f θ(g) + θ(f) g.
If S, T are unital associative algebras over a ring R, then an algebra homomorphism from S to T is a ring homomorphism that is also an R-module homomorphism.
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