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 ...

, especially in the area of

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 ...

known as

module theory 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 ...

, a principal indecomposable module has many important relations to the study of 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 ...

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 ...

, especially its

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 cycl ...

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 free modules. Various equivalent characterizati ...

s, and

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. Jacobson (2009), p. 111. Indecomposable is a weaker notion than simple module (which is also sometimes called irredu ...

Definition

A (left) principal indecomposable module of a ring ''R'' is a (left)

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 ...

of ''R'' that is a

direct summand 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 ...

of ''R'' and is an

. Alternatively, it is an indecomposable, projective,

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 cyclic group, that is, an Abelian group (i.e. Z-mod ...

. 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 or even

semiperfect In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. ...

, 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 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 L ...

''P'', which is a PIM, being an indecomposable, projective, cyclic module. Similarly over a

semiperfect ring In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exi ...

, every indecomposable projective module is a PIM, and every finitely generated projective module is a direct sum of PIMs. In the context of group algebras of

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

s over

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 ...

(which are semiperfect rings), the

representation ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representati ...

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 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 ...

that is not a PID, the

ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...

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.

References

* * * * * *{{Citation , last1=Nagao , first1=Hirosi , last2=Tsushima , first2=Yukio , title=Representations of finite groups , publisher=

Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference ...

, location=Boston, MA , isbn=978-0-12-513660-0 , mr=998775 , year=1989 , url-access=registration , url=https://archive.org/details/representationso0000naga Representation theory of finite groups Module theory