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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the isomorphism theorems (also known as Noether's isomorphism theorems) are
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s that describe the relationship between quotients,
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s, and
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,M ...
s. Versions of the theorems exist for
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
,
rings 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 ...
,
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,
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
,
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, and various other
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s. In
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
, the isomorphism theorems can be generalized to the context of algebras and congruences.


History

The isomorphism theorems were formulated in some generality for homomorphisms of modules by
Emmy Noether Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ...
in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
. Less general versions of these theorems can be found in work of
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
and previous papers by Noether. Three years later, B.L. van der Waerden published his influential ''
Moderne Algebra ''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', th ...
'' the first
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
textbook that took the
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
-
rings 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 ...
-
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
approach to the subject. Van der Waerden credited lectures by Noether on
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
on algebra, as well as a seminar conducted by Artin,
Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taught ...
,
Otto Schreier Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. Life His parents were the arc ...
, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called ''homomorphism theorem'', and ''two laws of isomorphism'' when applied to groups, appear explicitly.


Groups

We first present the isomorphism theorems of the
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
.


Note on numbers and names

Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules. It is less common to include the Theorem D, usually known as the ''
lattice theorem In group theory, the correspondence theorem (also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism theorem ) states that if N is a normal subgroup of ...
'' or the ''correspondence theorem'', as one of isomorphism theorems, but when included, it is the last one.


Statement of the theorems


Theorem A (groups)

Let ''G'' and ''H'' be groups, and let ''f'' : ''G'' → ''H'' be a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
. Then: # 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 learnin ...
of ''f'' is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of ''G'', # The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''f'' is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of ''H'', and # The image of ''f'' is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
''G'' / ker(''f''). In particular, if ''f'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
then ''H'' is isomorphic to ''G'' / ker(''f'').


Theorem B (groups)

Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then the following hold: # The
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
SN is a subgroup of G, # The
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
S \cap N is a normal subgroup of S, and # The quotient groups (SN)/N and S/(S\cap N) are isomorphic. Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the
normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of N in G. In this case, the intersection S \cap N is not a normal subgroup of G, but it is still a normal subgroup of S. This theorem is sometimes called the ''isomorphism theorem'', ''diamond theorem'' or the ''parallelogram theorem''. An application of the second isomorphism theorem identifies
projective linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
s: for example, the group on the
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
starts with setting G = \operatorname_2(\mathbb), the group of
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
2 × 2
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, S = \operatorname_2(\mathbb), the subgroup of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
1 matrices, and N the normal subgroup of scalar matrices \mathbb^\!I = \left\, we have S \cap N = \, where I is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, and SN = \operatorname_2(\mathbb). Then the second isomorphism theorem states that: : \operatorname_2(\mathbb) := \operatorname_2 \left(\mathbb)/(\mathbb^\!I\right) \cong \operatorname_2(\mathbb)/\ =: \operatorname_2(\mathbb)


Theorem C (groups)

Let G be a group, and N a normal subgroup of G. Then # If K is a subgroup of G such that N \subseteq K \subseteq G, then G/N has a subgroup isomorphic to K/N. # Every subgroup of G/N is of the form K/N for some subgroup K of G such that N \subseteq K \subseteq G. # If K is a normal subgroup of G such that N \subseteq K \subseteq G, then G/N has a normal subgroup isomorphic to K/N. # Every normal subgroup of G/N is of the form K/N for some normal subgroup K of G such that N \subseteq K \subseteq G. # If K is a normal subgroup of G such that N \subseteq K \subseteq G, then the quotient group (G/N)/(K/N) is isomorphic to G/K.


Theorem D (groups)

The correspondence theorem (also known as the lattice theorem) is sometimes called the third or fourth isomorphism theorem. The
Zassenhaus lemma Zassenhaus is a German surname. Notable people with the surname include: * Hans Zassenhaus (1912–1991), German mathematician ** Zassenhaus algorithm ** Zassenhaus group ** Zassenhaus lemma * Hiltgunt Zassenhaus (1916–2004), German philologi ...
(also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.


Discussion

The first isomorphism theorem can be expressed in category theoretical language by saying that the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the
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 ...
s form a factorization system for the
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. This is captured in the
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
in the margin, which shows the
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
and
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s whose existence can be deduced from the morphism f : G \rightarrow H. The diagram shows that every morphism in the category of groups has a
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 learnin ...
in the category theoretical sense; the arbitrary morphism ''f'' factors into \iota \circ \pi, where ''ι'' is a monomorphism and ''π'' is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object \ker f and a monomorphism \kappa: \ker f \rightarrow G (kernels are always monomorphisms), which complete the
short 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 o ...
running from the lower left to the upper right of the diagram. The use of the
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 o ...
convention saves us from having to draw the
zero morphism In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Definitions Suppose C is a category, and ''f'' : ''X'' → ''Y'' is a morphism in C. The ...
s from \ker f to H and G / \ker f. If the sequence is right split (i.e., there is a morphism ''σ'' that maps G / \operatorname f to a -preimage of itself), then ''G'' is the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
of the normal subgroup \operatorname \kappa and the subgroup \operatorname \sigma. If it is left split (i.e., there exists some \rho: G \rightarrow \operatorname f such that \rho \circ \kappa = \operatorname_), then it must also be right split, and \operatorname \kappa \times \operatorname \sigma is a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
decomposition of ''G''. In general, the existence of a right split does not imply the existence of a left split; but in an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
(such as that of abelian groups), left splits and right splits are equivalent by the
splitting lemma In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence : 0 \longrightarrow A \mathrel B \mathrel C \longrightarrow ...
, and a right split is sufficient to produce a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
decomposition \operatorname \kappa \oplus \operatorname \sigma. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence 0 \rightarrow G / \operatorname f \rightarrow H \rightarrow \operatorname f \rightarrow 0. In the second isomorphism theorem, the product ''SN'' is the
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...
of ''S'' and ''N'' in the
lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, a ...
of ''G'', while the intersection ''S'' ∩ ''N'' is the
meet Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
. The third isomorphism theorem is generalized by the
nine lemma right In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all co ...
to
abelian categories In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
and more general maps between objects.


Rings

The statements of the theorems for
rings 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 ...
are similar, with the notion of a normal subgroup replaced by the notion of an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
.


Theorem A (rings)

Let ''R'' and ''S'' be rings, and let ''φ'' : ''R'' → ''S'' be 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 preservi ...
. Then: # 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 learnin ...
of ''φ'' is an ideal of ''R'', # The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''φ'' is a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
of ''S'', and # The image of ''φ'' is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
''R'' / ker(''φ''). In particular, if ''φ'' is surjective then ''S'' is isomorphic to ''R'' / ker(''φ'').


Theorem B (rings)

Let ''R'' be a ring. Let ''S'' be a subring of ''R'', and let ''I'' be an ideal of ''R''. Then: # The sum ''S'' + ''I'' =  is a subring of ''R'', # The intersection ''S'' ∩ ''I'' is an ideal of ''S'', and # The quotient rings (''S'' + ''I'') / ''I'' and ''S'' / (''S'' ∩ ''I'') are isomorphic.


Theorem C (rings)

Let ''R'' be a ring, and ''I'' an ideal of ''R''. Then # If A is a subring of R such that I \subseteq A \subseteq R, then A/I is a subring of R/I. # Every subring of R/I is of the form A/I for some subring A of R such that I \subseteq A \subseteq R. # If J is an ideal of R such that I \subseteq J \subseteq R, then J/I is an ideal of R/I. # Every ideal of R/I is of the form J/I for some ideal J of R such that I \subseteq J \subseteq R. # If J is an ideal of R such that I \subseteq J \subseteq R, then the quotient ring (R/I)/(J/I) is isomorphic to R/J.


Theorem D (rings)

Let I be an ideal of R. The correspondence A\leftrightarrow A/I is an
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
-preserving
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between the set of subrings A of R that contain I and the set of subrings of R/I. Furthermore, A (a subring containing I) is an ideal of R
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
A/I is an ideal of R/I.


Modules

The statements of the isomorphism theorems for
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
are particularly simple, since it is possible to form a
quotient module In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by t ...
from any
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 mod ...
. The isomorphism theorems for
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 (modules over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
) and
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 commut ...
s (modules over \mathbb) are special cases of these. For
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
vector spaces, all of these theorems follow from the
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theor ...
. In the following, "module" will mean "''R''-module" for some fixed ring ''R''.


Theorem A (modules)

Let ''M'' and ''N'' be modules, and let ''φ'' : ''M'' → ''N'' be a
module homomorphism 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 f: M \to N is called an ''R''-''module homomorphism'' or an '' ...
. Then: # 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 learnin ...
of ''φ'' is a submodule of ''M'', # The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''φ'' is a submodule of ''N'', and # The image of ''φ'' is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
quotient module In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by t ...
''M'' / ker(''φ''). In particular, if ''φ'' is surjective then ''N'' is isomorphic to ''M'' / ker(''φ'').


Theorem B (modules)

Let ''M'' be a module, and let ''S'' and ''T'' be submodules of ''M''. Then: # The sum ''S'' + ''T'' =  is a submodule of ''M'', # The intersection ''S'' ∩ ''T'' is a submodule of ''M'', and # The quotient modules (''S'' + ''T'') / ''T'' and ''S'' / (''S'' ∩ ''T'') are isomorphic.


Theorem C (modules)

Let ''M'' be a module, ''T'' a submodule of ''M''. # If S is a submodule of M such that T \subseteq S \subseteq M, then S/T is a submodule of M/T. # Every submodule of M/T is of the form S/T for some submodule S of M such that T \subseteq S \subseteq M. # If S is a submodule of M such that T \subseteq S \subseteq M, then the quotient module (M/T)/(S/T) is isomorphic to M/S.


Theorem D (modules)

Let M be a module, N a submodule of M. There is a bijection between the submodules of M that contain N and the submodules of M/N. The correspondence is given by A\leftrightarrow A/N for all A\supseteq N. This correspondence commutes with the processes of taking sums and intersections (i.e., is a
lattice isomorphism A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
between the lattice of submodules of M/N and the lattice of submodules of M that contain N).


Universal algebra

To generalise this to
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
, normal subgroups need to be replaced by
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
s. A congruence on an
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 ...
A is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
\Phi\subseteq A \times A that forms a subalgebra of A \times A considered as an algebra with componentwise operations. One can make the set of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es A/\Phi into an algebra of the same type by defining the operations via representatives; this will be
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A funct ...
since \Phi is a subalgebra of A \times A. The resulting structure is the quotient algebra.


Theorem A (universal algebra)

Let f:A \rightarrow B be an algebra
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
. Then the image of f is a subalgebra of B, the relation given by \Phi:f(x)=f(y) (i.e. 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 learnin ...
of f) is a congruence on A, and the algebras A/\Phi and \operatorname f are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. (Note that in the case of a group, f(x)=f(y)
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...
f(xy^) = 1, so one recovers the notion of kernel used in group theory in this case.)


Theorem B (universal algebra)

Given an algebra A, a subalgebra B of A, and a congruence \Phi on A, let \Phi_B = \Phi \cap (B \times B) be the trace of \Phi in B and \Phi=\ the collection of equivalence classes that intersect B. Then # \Phi_B is a congruence on B, # \ \Phi is a subalgebra of A/\Phi, and # the algebra \Phi is isomorphic to the algebra B/\Phi_B.


Theorem C (universal algebra)

Let A be an algebra and \Phi, \Psi two congruence relations on A such that \Psi \subseteq \Phi. Then \Phi/\Psi = \ = \Psi \circ \Phi \circ \Psi^ is a congruence on A/\Psi, and A/\Phi is isomorphic to (A/\Psi)/(\Phi/\Psi).


Theorem D (universal algebra)

Let A be an algebra and denote \operatornameA the set of all congruences on A. The set \operatornameA is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
ordered by inclusion. If \Phi\in\operatornameA is a congruence and we denote by \left Phi,A\times A\rightsubseteq\operatornameA the set of all congruences that contain \Phi (i.e. \left Phi,A\times A\right/math> is a principal
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
in \operatornameA, moreover it is a sublattice), then the map \alpha:\left Phi,A\times A\rightto\operatorname(A/\Phi),\Psi\mapsto\Psi/\Phi is a lattice isomorphism.


Note


References

*
Emmy Noether Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ...
, ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'',
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
96 (1927) pp. 26–61 * Colin McLarty, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". ''The Architecture of Modern Mathematics: Essays in history and philosophy'' (edited by
Jeremy Gray Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics. Biography Gray studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his Ph.D ...
and José Ferreirós), Oxford University Press (2006) pp. 211–35. * * Paul M. Cohn, ''Universal algebra'', Chapter II.3 p. 57 * * * * * * * * * {{citation , author = Joseph J. Rotman , title=Advanced Modern Algebra , publisher=Prentice Hall , edition= 2 , year=2003 , isbn=0130878685