# Module homomorphism

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{{short description|Linear map over a ring}}In [algebra](/source/Abstract_algebra), a '''module homomorphism''' is a [function](/source/function_(mathematics)) between [module](/source/module_(mathematics))s that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a [ring](/source/Ring_(mathematics)) ''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](/source/group_homomorphism) (for the underlying additive groups) that commutes with [scalar multiplication](/source/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](/source/preimage) of the zero element under ''f'' is called the [kernel](/source/kernel_(algebra)) of ''f''. The [set](/source/Set_(mathematics)) of all module homomorphisms from ''M'' to ''N'' is denoted by <math>\operatorname{Hom}_R(M, N)</math>. It is an [abelian group](/source/abelian_group) (under pointwise addition) but is not necessarily a module unless ''R'' is [commutative](/source/Commutative_ring).

The [composition](/source/Function_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](/source/category_of_modules).

== Terminology ==
A module homomorphism is called a ''module isomorphism'' if it admits an inverse homomorphism; in particular, it is a [bijection](/source/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](/source/if_and_only_if) it is an isomorphism between the underlying abelian groups.

The [isomorphism theorem](/source/isomorphism_theorem)s hold for module homomorphisms.

A module homomorphism from a module ''M'' to itself is called an [endomorphism](/source/endomorphism) and an isomorphism from ''M'' to itself an [automorphism](/source/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](/source/endomorphism_ring) of ''M''. The [group of units](/source/group_of_units) of this ring is the [automorphism group](/source/automorphism_group) of ''M''. 

[Schur's lemma](/source/Schur's_lemma) says that a homomorphism between [simple module](/source/simple_module)s (modules with no non-trivial [submodule](/source/submodule)s) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a [division ring](/source/division_ring).

In the language of the [category theory](/source/category_theory), an injective homomorphism is also called a [monomorphism](/source/monomorphism) and a surjective homomorphism an [epimorphism](/source/epimorphism).

== Examples ==
*The [zero map](/source/zero_map) ''M'' → ''N'' that maps every element to zero.
*A [linear transformation](/source/linear_transformation) between [vector space](/source/vector_space)s.
*<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](/source/Ideal_(ring_theory)) ''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](/source/annihilator_(ring_theory)) 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](/source/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''.<ref name=bourbaki/> (The module structure on Hom here comes from the right ''R''-action on ''R''; see #Module structures on Hom below.)
**<math>\operatorname{Hom}_R(M, R)</math> is called the [dual module](/source/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 <math>M^*</math>.
*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](/source/derivation_(algebra)) if for any ''f'', ''g'' in ''S'', {{nowrap|θ(''f g'') <nowiki>=</nowiki> ''f'' θ(''g'') + θ(''f'') ''g''}}.
*If ''S'', ''T'' are unital [associative algebra](/source/associative_algebra)s over a ring ''R'', then an [algebra homomorphism](/source/algebra_homomorphism) from ''S'' to ''T'' is a [ring homomorphism](/source/ring_homomorphism) that is also an ''R''-module homomorphism.

== Module structures on Hom ==
In short, Hom inherits a ring action that was not ''used up'' to form Hom. More precise, let ''M'', ''N'' be left ''R''-modules. Suppose ''M'' has a right action of a ring ''S'' that commutes with the ''R''-action; i.e., ''M'' is an (''R'', ''S'')-module. Then
:<math>\operatorname{Hom}_R(M, N)</math>
has the structure of a left ''S''-module defined by: for ''s'' in ''S'' and ''x'' in ''M'',
:<math>(s \cdot f)(x) = f(xs).</math>
It is well-defined (i.e., <math>s \cdot f</math> is ''R''-linear) since
:<math>(s \cdot f)(rx) = f(rxs) = rf(xs) = r (s \cdot f)(x),</math>
and <math>s \cdot f</math> is a ring action since
:<math>(st \cdot f)(x) = f(xst) = (t \cdot f)(xs) = s \cdot (t \cdot f)(x)</math>.

Note: the above verification would "fail" if one used the left ''R''-action in place of the right ''S''-action. In this sense, Hom is often said to "use up" the ''R''-action.

Similarly, if ''M'' is a left ''R''-module and ''N'' is an (''R'', ''S'')-module, then <math>\operatorname{Hom}_R(M, N)</math> is a right ''S''-module by <math>(f \cdot s)(x) = f(x)s</math>.

== A matrix representation ==
The relationship between matrices and linear transformations in [linear algebra](/source/linear_algebra) generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right ''R''-module ''U'', there is the [canonical isomorphism](/source/canonical_isomorphism) of the abelian groups
:<math>\operatorname{Hom}_R(U^{\oplus n}, U^{\oplus m}) \overset{f \mapsto [f_{ij}]}\underset{\sim}\to M_{m, n}(\operatorname{End}_R(U))</math>
obtained by viewing <math>U^{\oplus n}</math> consisting of column vectors and then writing ''f'' as an ''m'' × ''n'' matrix. In particular, viewing ''R'' as a right ''R''-module and using <math>\operatorname{End}_R(R) \simeq R</math>, one has
:<math>\operatorname{End}_R(R^n) \simeq M_n(R)</math>,
which turns out to be a ring isomorphism (as a composition corresponds to a [matrix multiplication](/source/matrix_multiplication)).

Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rank [free module](/source/free_module)s, then a choice of an ordered basis corresponds to a choice of an isomorphism <math>F \simeq R^n</math>. The above procedure then gives the matrix representation with respect to such choices of the bases. For more general modules, matrix representations may either lack uniqueness or not exist.

== Defining ==
In practice, one often defines a module homomorphism by specifying its values on a [generating set](/source/generating_set_of_a_module). More precisely, let ''M'' and ''N'' be left ''R''-modules. Suppose a [subset](/source/subset) ''S'' generates ''M''; i.e., there is a surjection <math>F \to M</math> with a free module ''F'' with a basis indexed by ''S'' and kernel ''K'' (i.e., one has a [free presentation](/source/free_presentation)). Then to give a module homomorphism <math>M \to N</math> is to give a module homomorphism <math>F \to N</math> that kills ''K'' (i.e., maps ''K'' to zero).

== Operations ==
If <math>f: M \to N</math> and <math>g: M' \to N'</math> are module homomorphisms, then their direct sum is
:<math>f \oplus g: M \oplus M' \to N \oplus N', \, (x, y) \mapsto (f(x), g(y))</math>
and their tensor product is
:<math>f \otimes g: M \otimes M' \to N \otimes N', \, x \otimes y \mapsto f(x) \otimes g(y).</math>

Let <math>f: M \to N</math> be a module homomorphism between left modules. The [graph](/source/graph_of_a_function) Γ<sub>''f''</sub> of ''f'' is the submodule of ''M'' ⊕ ''N'' given by
:<math>\Gamma_f = \{ (x, f(x)) | x \in M \}</math>,
which is the image of the module homomorphism {{nowrap|''M'' → ''M'' ⊕ ''N'', ''x'' → (''x'', ''f''(''x'')), called the '''graph morphism'''.<!-- how to write mapsto in html? -->}}

The [transpose](/source/transpose) of ''f'' is
:<math>f^*: N^* \to M^*, \, f^*(\alpha) = \alpha \circ f.</math>
If ''f'' is an isomorphism, then the transpose of the inverse of ''f'' is called the '''contragredient''' of ''f''.

== Exact sequences ==
Consider a sequence of module homomorphisms
:<math>\cdots \overset{f_3}\longrightarrow M_2 \overset{f_2}\longrightarrow M_1 \overset{f_1}\longrightarrow M_0 \overset{f_0}\longrightarrow M_{-1} \overset{f_{-1}}\longrightarrow \cdots.</math>
Such a sequence is called a [chain complex](/source/chain_complex) (or often just complex) if each composition is zero; i.e., <math>f_i \circ f_{i+1} = 0</math> or equivalently the image of <math>f_{i+1}</math> is contained in the kernel of <math>f_i</math>. (If the numbers increase instead of decrease, then it is called a cochain complex; e.g., [de Rham complex](/source/de_Rham_complex).) A chain complex is called an [exact sequence](/source/exact_sequence) if <math>\operatorname{im}(f_{i+1}) = \operatorname{ker}(f_i)</math>. A special case of an exact sequence is a short exact sequence:
:<math>0 \to A \overset{f}\to B \overset{g}\to C \to 0</math>
where <math>f</math> is injective, the kernel of <math>g</math> is the image of <math>f</math> and <math>g</math> is surjective.

Any module homomorphism <math>f : M \to N</math> defines an exact sequence
:<math>0 \to K \to M \overset{f}\to N \to C \to 0,</math>
where <math>K</math> is the kernel of <math>f</math>, and <math>C</math> is the [cokernel](/source/cokernel), that is the quotient of <math>N</math> by the image of <math>f</math>.

In the case of modules over a [commutative ring](/source/commutative_ring), a sequence is exact if and only if it is exact at all the [maximal ideal](/source/maximal_ideal)s; that is all sequences 
:<math>0 \to A_{\mathfrak{m}} \overset{f}\to B_{\mathfrak{m}} \overset{g}\to C_{\mathfrak{m}} \to 0</math>
are exact, where the subscript <math>{\mathfrak{m}}</math> means the [localization](/source/localization_of_a_module) at a maximal ideal <math>{\mathfrak{m}}</math>.

If <math>f : M \to B, g: N \to B</math> are module homomorphisms, then they are said to form a '''fiber square''' (or '''[pullback square](/source/pullback_square)'''), denoted by ''M'' ×<sub>''B''</sub> ''N'', if it fits into
:<math>0 \to M \times_{B} N \to M \times N \overset{\phi}\to B \to 0</math>
where <math>\phi(x, y) = f(x) - g(x)</math>.

Example: Let <math>B \subset A</math> be commutative rings, and let ''I'' be the [annihilator](/source/annihilator_(ring_theory)) of the quotient ''B''-module ''A''/''B'' (which is an ideal of ''A''). Then canonical maps <math>A \to A/I, B/I \to A/I</math> form a fiber square with <math>B = A \times_{A/I} B/I.</math>

== Endomorphisms of finitely generated modules ==
Let <math>\phi: M \to M</math> be an endomorphism between finitely generated ''R''-modules for a commutative ring ''R''. Then
*<math>\phi</math> is killed by its characteristic polynomial relative to the generators of ''M''; see [Nakayama's lemma#Proof](/source/Nakayama's_lemma).
*If <math>\phi</math> is surjective, then it is injective.<ref name=matsumura/>

See also: [Herbrand quotient](/source/Herbrand_quotient) (which can be defined for any endomorphism with some finiteness conditions.)

== Variant: additive relations ==
{{see also|binary relation}}
An '''additive relation''' <math>M \to N</math> from a module ''M'' to a module ''N'' is a submodule of <math>M \oplus N.</math><ref name=maclane/> In other words, it is a "[many-valued](/source/many-valued_function)" homomorphism defined on some submodule of ''M''. The inverse <math>f^{-1}</math> of ''f'' is the submodule <math>\{ (y, x) | (x, y) \in f \}</math>. Any additive relation ''f'' determines a homomorphism from a submodule of ''M'' to a quotient of ''N''
:<math>D(f) \to N/\{ y | (0, y) \in f \}</math>
where <math>D(f)</math> consists of all elements ''x'' in ''M'' such that (''x'', ''y'') belongs to ''f'' for some ''y'' in ''N''.

A [transgression](/source/Spectral_sequence) that arises from a spectral sequence is an example of an additive relation.

== See also ==
*[Mapping cone (homological algebra)](/source/Mapping_cone_(homological_algebra))
*[Smith normal form](/source/Smith_normal_form)
*[Chain complex](/source/Chain_complex)
*[Pairing](/source/Pairing)

== Notes ==
<references>

<ref name=bourbaki>{{citation
 | last = Bourbaki | first = Nicolas | author-link = Nicolas Bourbaki
 | contribution = Chapter II, §1.14, remark 2
 | isbn = 3-540-64243-9
 | mr = 1727844
 | publisher = Springer-Verlag
 | series = Elements of Mathematics
 | title = Algebra I, Chapters 1–3
 | year = 1998}}</ref>

<ref name=maclane>{{citation
 | last = Mac Lane | first = Saunders | author-link = Saunders Mac Lane
 | isbn = 3-540-58662-8
 | mr = 1344215
 | page = [https://books.google.com/books?id=ujRqCQAAQBAJ&pg=PA52 52]
 | publisher = Springer-Verlag
 | series = Classics in Mathematics
 | title = Homology
 | year = 1995}}</ref>

<ref name=matsumura>{{citation
 | last = Matsumura | first = Hideyuki
 | contribution = Theorem 2.4
 | edition = 2nd
 | isbn = 0-521-36764-6
 | mr = 1011461
 | publisher = Cambridge University Press
 | series = Cambridge Studies in Advanced Mathematics
 | title = Commutative Ring Theory
 | volume = 8
 | year = 1989}}</ref>

</references>

Category:Algebra
Category:Module theory

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Adapted from the Wikipedia article [Module homomorphism](https://en.wikipedia.org/wiki/Module_homomorphism) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Module_homomorphism?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.