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{{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]].

== 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 theorem]]s 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 module]]s (modules with no non-trivial [[submodule]]s) 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 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 [[Ideal (ring theory)|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 (ring theory)|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''.<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]] 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 (algebra)|derivation]] 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]]s over a ring ''R'', then an [[algebra homomorphism]] from ''S'' to ''T'' is a [[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]] generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right ''R''-module ''U'', there is the [[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]]).

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]]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 of a module|generating set]]. More precisely, let ''M'' and ''N'' be left ''R''-modules. Suppose a [[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]]). 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 of a function|graph]] 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'''.}}

The [[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]] (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]].) A chain complex is called an [[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]], that is the quotient of <math>N</math> by the image of <math>f</math>.

In the case of modules over a [[commutative ring]], a sequence is exact if and only if it is exact at all the [[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 of a module|localization]] 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]]'''), 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 (ring theory)|annihilator]] 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]].
*If <math>\phi</math> is surjective, then it is injective.<ref name=matsumura/>

See also: [[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 function|many-valued]]" 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 [[Spectral sequence#Edge maps and transgressions|transgression]] that arises from a spectral sequence is an example of an additive relation.

== See also ==
*[[Mapping cone (homological algebra)]]
*[[Smith normal form]]
*[[Chain complex]]
*[[Pairing]]

== Notes ==
{{reflist|refs=

<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>

}}

[[Category:Algebra]]
[[Category:Module theory]]

Module homomorphism - Revision history

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