Module homomorphism

{{short description|Linear map over a ring}}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''.<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 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 algebras 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 modules, 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. 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 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 ideals; 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 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 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" 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 that arises from a spectral sequence is an example of an additive relation.

== 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