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 <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 of f. The 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.
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 <math>\operatorname{End}_R(M) = \operatorname{Hom}_R(M, M)</math> 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.
- <math>\operatorname{Hom}_{\mathbb{Z(\mathbb{Z}/n, \mathbb{Z}/m) = \mathbb{Z}/\operatorname{gcd}(n,m)</math>.
- For a commutative ring R and ideals I, J, there is the canonical identification
- :<math>\operatorname{Hom}_R(R/I, R/J) = \{ r \in R | r I \subset J \}/J</math>
:given by <math>f \mapsto f(1)</math>. In particular, <math>\operatorname{Hom}_R(R/I, R)</math> is the annihilator of I.
- Given a ring R and an element r, let <math>l_r: R \to R</math> denote the left multiplication by r. Then for any s, t in R,
- :<math>l_r(st) = rst = l_r(s)t</math>.
:That is, <math>l_r</math> is right R-linear.
- For any ring R,
- <math>\operatorname{End}_R(R) = R</math> as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation <math>R \overset{\sim}\to \operatorname{End}_R(R), \, r \mapsto l_r</math>.
- Similarly, <math>\operatorname{End}_R(R) = R^{op}</math> as rings when R is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
- <math>\operatorname{Hom}_R(R, M) = M</math> through <math>f \mapsto f(1)</math> for any left module M.
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