Krull–Schmidt theorem
In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable su
Associated prime
In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually de
Noetherian module
In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is an
Bimodule
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of ma
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 q
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
Pure submodule
In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules
Principal indecomposable module
In mathematics, especially in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simpl
Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebra
Cyclic module
In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cycl
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.
Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be
Dense submodule
In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If N is a dense submodule of M, it may alternatively be said
Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent (denoted by ) if
Lattice (module)
In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedd
Radical of a module
In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways,
Balanced module
In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian gro
Module (mathematics)
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, sinc
Uniform module
In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an essential su
Resolution (algebra)
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objec
Support of a module
In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals of A such that (that is, the localization of M at is not equal to zero). It is denoted by . T
Isotypical representation
In group theory, an isotypical, primary or factor representation of a group G is a unitary representation such that any two subrepresentations have equivalent sub-subrepresentations. This is related t
Primitive ideal
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are
Fitting lemma
The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every
Decomposition of a module
In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisi
Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Indecomposable is a weaker notion than simple module (which is also
Cosocle
In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal norma
Free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coeffic
Quasi-Frobenius ring
In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that o
Essential extension
In mathematics, specifically module theory, given a ring R and an R-module M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or la
Continuous module
In mathematics, a continuous module is a module M such that every submodule of M is essential in a direct summand and every submodule of M isomorphic to a direct summand is itself a direct summand. Th
Comodule
In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebr
Serial module
In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A modu
Artin–Rees lemma
In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by th
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which g
Glossary of module theory
Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject. See also: Glossary of linear algebra, Glossary of ring theory, Glossary of rep
Invariant basis number
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the
Global dimension
In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or i
Modular representation theory
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, nec
Flat cover
In algebra, a flat cover of a module M over a ring is a surjective homomorphism from a flat module F to M that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (n
Character module
In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the o
Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the
Injective hull
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective
Torsionless module
In abstract algebra, a module M over a ring R is called torsionless if it can be embedded into some direct product RI. Equivalently, M is torsionless if each non-zero element of M has non-zero image u
Semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.
Algebraically compact module
In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the mod
Depth (ring theory)
In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over
Projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of
Elementary divisors
In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If is a PID
Flat module
In algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a se
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
Localization (commutative algebra)
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/
Stably free module
In mathematics, a stably free module is a module which is close to being free.
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 endom
Projective cover
In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of in
Supermodule
In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetr
Kaplansky's theorem on projective modules
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is c
Length of a module
In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational number
Annihilator (ring theory)
In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of S. Over an integral domain,
Invariant factor
The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If is a PID and a finitel
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as sub
Eilenberg–Mazur swindle
In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums. In geometric topology it was intr
Semimodule
In mathematics, a semimodule over a semiring R is like a module over a ring except that it is only a commutative monoid rather than an abelian group.
Countably generated module
In mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes from Kaplansky's theo
Module of covariants
In algebra, given an algebraic group G, a G-module M and a G-algebra A, all over a field k, the module of covariants of type M is the -module where refers to taking the elements fixed by the action of
Hopfian object
In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism. The dual notion is that of a cohopfian object
Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context
Algebra representation
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be ma
Singular submodule
In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (r
Schanuel's lemma
In mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is usef
Artinian module
In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for
Mitchell's embedding theorem
Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather ab
Top (algebra)
In the context of a module M over a ring R, the top of M is the largest semisimple quotient module of M if it exists. For finite-dimensional k-algebras (k a field) R, if rad(M) denotes the intersectio
Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of