Krull–Schmidt theorem

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...

, the Krull–Schmidt theorem states that a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

subjected to certain finiteness conditions on chains of subgroup

Subgroup

In group theory, 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 G if the restriction of * to H x H is a group operation on H...

s, can be uniquely written as a finite direct product

Direct product of groups

In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

of indecomposable subgroups.

Definitions

We say that a group G satisfies the ascending chain condition

Ascending chain condition

The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...

(ACC) on subgroups if every sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

of subgroups of G:

is eventually constant, i.e., there exists N such that G_N = G_N+1 = G_N+2 = ... . We say that G satisfies the ACC on normal subgroups if every such sequence of normal subgroups of G eventually becomes constant.

Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups:

Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group

satisfies ACC but not DCC, since (2) > (2)² > (2)³ > ... is an infinite decreasing sequence of subgroups. On the other hand, the

-torsion part of

(the quasicyclic p-group

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...

) satisfies DCC but not ACC.

We say a group G is indecomposable if it cannot be written as a direct product of non-trivial subgroups G = H × K.

The theorem says:

If

is a group that satisfies ACC and DCC on normal subgroups, then there is a unique way of writing

as a direct product

of finitely many indecomposable subgroups of

. Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property. That is: suppose

is another expression of

as a product of indecomposable subgroups. Then

and there is a reindexing of the

's satisfying

and are isomorphic for each ;
for each .

Krull–Schmidt theorem for modules

is a module

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...

that satisfies the ACC and DCC on submodules (that is, it is both Noetherian

Noetherian module

In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....

and Artinian

Artinian module

In 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 rings, and a ring is Artinian if and only if it is an Artinian module over itself...

or – equivalently – of finite length

Length of a module

In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...

), then

is a direct sum

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...

of 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. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.

In general, the theorem fails, if one only assumes that the module is Noetherian.

History

The present-day Krull–Schmidt theorem was first proved by Joseph Wedderburn

Joseph Wedderburn

Joseph Henry Maclagan Wedderburn was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras...

(Ann. of Math (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A. Miller

George Abram Miller

George Abram Miller was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance...

where direct products of abelian groups were considered. Wedderburn's theorem is stated as an exchange property between direct decompositions of maximum length. However, Wedderburn's proof makes no use of automorphisms.

The thesis of Robert Remak

Robert Remak (mathematician)

Robert Erich Remak was a German mathematician. He is chiefly remembered for his work in group theory . His other interests included algebraic number theory, mathematical economics and geometry of numbers...

(1911) derived the same uniqueness result as Wedderburn but also proved (in modern terminology) that the group of central automorphisms acts transitively on the set of direct decompositions of maximum length of a finite group. From that stronger theorem Remak also proved various corollaries including that groups with a trivial center and perfect groups have a unique Remak decomposition.

Otto Schmidt

Otto Schmidt

Otto Yulyevich Schmidt was a Soviet scientist, mathematician, astronomer, geophysicist, statesman, academician, Hero of the USSR , and member of the Communist Party.-Biography:He was born in Mogilev, Russian Empire...

(Sur les produits directs, S. M. F. Bull. 41 (1913), 161–164), simplified the main theorems of Remak to the 3 page predecessor to today's textbook proofs. His method improves Remak's use of idempotents to create the appropriate central automorphisms. Both Remak and Schmidt published subsequent proofs and corollaries to their theorems.

Wolfgang Krull

Wolfgang Krull

Wolfgang Krull was a German mathematician working in the field of commutative algebra.He was born in Baden-Baden, Imperial Germany and died in Bonn, West Germany.- See also :* Krull dimension* Krull topology...

(Über verallgemeinerte endliche Abelsche Gruppen, M. Z. 23 (1925) 161–196), returned to G.A. Miller

George Abram Miller

George Abram Miller was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance...

's original problem of direct products of abelian groups by extending to abelian operator groups with ascending and descending chain conditions. This is most often stated in the language of modules. His proof observes that the idempotents used in the proofs of Remak and Schmidt can be restricted to module homomorphisms; the remaining details of the proof are largely unchanged.

O. Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions. This proof makes no use of idempotents and does not reprove the transitivity of Remak's theorems.

Kurosh's The Theory of Groups and Zassenhaus' The Theory of Groups include the proofs of Schmidt and Ore under the name of Remak–Schmidt but acknowledge Wedderburn and Ore. Later texts use the title Krull–Schmidt (Hungerford

Thomas W. Hungerford

Thomas William Hungerford is an American mathematician who works in algebra and mathematics education. He is the author or coauthor of several widely used and widely cited textbooks covering high-school to graduate-level mathematics. From 1963 until 1980 he taught at the University of Washington...

's Algebra) and Krull–Schmidt–Azumaya

Goro Azumaya

was a Japanese mathematician who introduced the notion of Azumaya algebra in 1951. His advisor was Shokichi Iyanaga. At the time of his death he was an emeritus professor at Indiana University.-External links:...

(Curtis–Reiner). The name Krull–Schmidt is now popularly substituted for any theorem concerning uniqueness of direct products of maximum size. Some authors choose to call direct decompositions of maximum-size Remak decompositions to honor his contributions.

External links

Page at PlanetMath

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Definitions

Krull–Schmidt theorem

Krull–Schmidt theorem for modules

History

Further reading

External links