Principal indecomposable module
Encyclopedia
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 simple module
s, projective module
s, and indecomposable module
s.
. Alternatively, it is an indecomposable, projective, cyclic module
. Principal indecomposable modules are also called PIMs for short.
If the ring R is Artinian
or even semiperfect
, then R is a direct sum of principal indecomposable modules, and there is one isomorphism class of PIM per isomorphism class of simple module. To each PIM P is associated its head, P/JP, which is a simple module, being an indecomposable semi-simple module. To each simple module S is associated its projective cover
P, which is a PIM, being an indecomposable, projective, cyclic module.
Similarly over a semiperfect ring
, every indecomposable projective module is a PIM, and every finitely generated projective module is a direct sum of PIMs.
In the context of group algebra
s of finite group
s over fields
(which are semiperfect rings), the representation ring
describes the indecomposable modules, and the modular characters
of simple modules represent both a subring and a quotient ring. The representation ring over the complex field is usually better understood and since PIMs correspond to modules over the complexes using p-modular system, one can use PIMs to transfer information from the complex representation ring to the representation ring over a field of positive characteristic. Roughly speaking this is called block theory.
Over a Dedekind domain
that is not a PID
, the ideal class group
measures the difference between projective indecomposable modules and principal indecomposable modules: the projective indecomposable modules are exactly the (modules isomorphic to) nonzero ideals and the principal indecomposable modules are precisely the (modules isomorphic to) nonzero principal ideals.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
known as module theory, a principal indecomposable module has many important relations to the study of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
's modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, especially its simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
s, projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s, and indecomposable module
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:simple means "no proper submodule" N...
s.
Definition
A (left) principal indecomposable module of a ring R is a (left) submodule of R that is a direct summand of R and is an indecomposable moduleIndecomposable 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:simple means "no proper submodule" N...
. Alternatively, it is an indecomposable, projective, cyclic module
Cyclic module
In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element.- Definition :...
. Principal indecomposable modules are also called PIMs for short.
Relations
The projective indecomposable modules over some rings have very close connections with those rings' simple, projective, and indecomposable modules.If the ring R is Artinian
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
or even semiperfect
Semiperfect ring
In abstract algebra, a semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left right symmetric.- Definition :Let R be ring...
, then R is a direct sum of principal indecomposable modules, and there is one isomorphism class of PIM per isomorphism class of simple module. To each PIM P is associated its head, P/JP, which is a simple module, being an indecomposable semi-simple module. To each simple module S is associated its projective cover
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 injective envelopes.- Definition :...
P, which is a PIM, being an indecomposable, projective, cyclic module.
Similarly over a semiperfect ring
Semiperfect ring
In abstract algebra, a semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left right symmetric.- Definition :Let R be ring...
, every indecomposable projective module is a PIM, and every finitely generated projective module is a direct sum of PIMs.
In the context of group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
s of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s over fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
(which are semiperfect rings), the representation ring
Representation ring
In mathematics, especially in the area of algebra known as representation theory, the representation ring of a group is a ring formed from all the linear representations of the group. For a given group, the ring will depend on the base field of the representations...
describes the indecomposable modules, and the modular characters
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
of simple modules represent both a subring and a quotient ring. The representation ring over the complex field is usually better understood and since PIMs correspond to modules over the complexes using p-modular system, one can use PIMs to transfer information from the complex representation ring to the representation ring over a field of positive characteristic. Roughly speaking this is called block theory.
Over a Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
that is not a PID
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, the ideal class group
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
measures the difference between projective indecomposable modules and principal indecomposable modules: the projective indecomposable modules are exactly the (modules isomorphic to) nonzero ideals and the principal indecomposable modules are precisely the (modules isomorphic to) nonzero principal ideals.