In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, a module homomorphism is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
s that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'', then a function
is called an ''R''-''module homomorphism'' or an ''R''-''linear map'' if for any ''x'', ''y'' in ''M'' and ''r'' in ''R'',
:
:
In other words, ''f'' is a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
(for the underlying additive groups) that commutes with scalar multiplication. If ''M'', ''N'' are right ''R''-modules, then the second condition is replaced with
:
The
preimage of the zero element under ''f'' is called the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
of ''f''. The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all module homomorphisms from ''M'' to ''N'' is denoted by
. It is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
(under pointwise addition) but is not necessarily a module unless ''R'' is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
.
The
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
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
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist ...
s hold for module homomorphisms.
A module homomorphism from a module ''M'' to itself is called an
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
and an isomorphism from ''M'' to itself an
automorphism. One writes
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
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
of ''M''. The
group of units of this ring is the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of ''M''.
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ' ...
says that a homomorphism between
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cy ...
s (modules with no non-trivial
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
s) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
.
In the language of the
category theory, an injective homomorphism is also called a
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
and a surjective homomorphism an
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
.
Examples
*The
zero map
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
''M'' → ''N'' that maps every element to zero.
*A
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
between
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s.
*
.
*For a commutative ring ''R'' and
ideals ''I'', ''J'', there is the canonical identification
*:
:given by
. In particular,
is the
annihilator of ''I''.
*Given a ring ''R'' and an element ''r'', let
denote the left multiplication by ''r''. Then for any ''s'', ''t'' in ''R'',
*:
.
:That is,
is ''right'' ''R''-linear.
*For any ring ''R'',
**
as rings when ''R'' is viewed as a right module over itself. Explicitly, this isomorphism is given by the
left regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular re ...
.
**Similarly,
as rings when ''R'' is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
**
through
for any left module ''M''. (The module structure on Hom here comes from the right ''R''-action on ''R''; see
#Module structures on Hom below.)
**
is called the
dual module In mathematics, the dual module of a left (respectively right) module ''M'' over a ring ''R'' is the set of module homomorphisms from ''M'' to ''R'' with the pointwise right (respectively left) module structure. The dual module is typically denote ...
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
.
*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
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
if for any ''f'', ''g'' in ''S'', .
*If ''S'', ''T'' are unital
associative algebras over a ring ''R'', then an
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF ...
from ''S'' to ''T'' is a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
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
:
has the structure of a left ''S''-module defined by: for ''s'' in ''S'' and ''x'' in ''M'',
:
It is well-defined (i.e.,
is ''R''-linear) since
:
and
is a ring action since
:
.
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
is a right ''S''-module by
.
A matrix representation
The relationship between matrices and linear transformations in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
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
:
obtained by viewing
consisting of column vectors and then writing ''f'' as an ''m'' × ''n'' matrix. In particular, viewing ''R'' as a right ''R''-module and using
, one has
:
,
which turns out to be a ring isomorphism (as a composition corresponds to a
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
).
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
. 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
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
. More precisely, let ''M'' and ''N'' be left ''R''-modules. Suppose a
subset ''S'' generates ''M''; i.e., there is a surjection
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
is to give a module homomorphism
that kills ''K'' (i.e., maps ''K'' to zero).
Operations
If
and
are module homomorphisms, then their direct sum is
:
and their tensor product is
:
Let
be a module homomorphism between left modules. The
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
Γ
''f'' of ''f'' is the submodule of ''M'' ⊕ ''N'' given by
:
,
which is the image of the module homomorphism
The
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of ''f'' is
:
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
:
Such a sequence is called a
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
(or often just complex) if each composition is zero; i.e.,
or equivalently the image of
is contained in the kernel of
. (If the numbers increase instead of decrease, then it is called a cochain complex; e.g.,
de Rham complex
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly ada ...
.) A chain complex is called an
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
if
. A special case of an exact sequence is a short exact sequence:
:
where
is injective, the kernel of
is the image of
and
is surjective.
Any module homomorphism
defines an exact sequence
:
where
is the kernel of
, and
is the cokernel, that is the quotient of
by the image of
.
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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
s; that is all sequences
:
are exact, where the subscript
means the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
at a maximal ideal
.
If
are module homomorphisms, then they are said to form a fiber square (or
pullback square), denoted by ''M'' ×
''B'' ''N'', if it fits into
:
where
.
Example: Let
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
form a fiber square with
Endomorphisms of finitely generated modules
Let
be an endomorphism between finitely generated ''R''-modules for a commutative ring ''R''. Then
*
is killed by its characteristic polynomial relative to the generators of ''M''; see
Nakayama's lemma#Proof.
*If
is surjective, then it is injective.
See also:
Herbrand quotient In mathematics, the Herbrand quotient is a quotient of orders of Group cohomology, cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.
Definition
If ''G'' is a finite cyc ...
(which can be defined for any endomorphism with some finiteness conditions.)
Variant: additive relations
An additive relation
from a module ''M'' to a module ''N'' is a submodule of
In other words, it is a "
many-valued" homomorphism defined on some submodule of ''M''. The inverse
of ''f'' is the submodule
. Any additive relation ''f'' determines a homomorphism from a submodule of ''M'' to a quotient of ''N''
:
where
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.
See also
*
Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain comp ...
*
Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...
*
Chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
*
Pairing
In mathematics, a pairing is an ''R''-bilinear map from the Cartesian product of two ''R''-modules, where the underlying ring ''R'' is commutative.
Definition
Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R''-mod ...
Notes
References
*Bourbaki, ''Algebra''. Chapter II.
*S. MacLane, ''Homology''{{full citation needed, date=July 2019
*H. Matsumura, ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
Algebra