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 semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module.

An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, is known as the Krull–Schmidt theorem.

A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a product) of matrix rings over division rings (this observation is known as the Artin–Wedderburn theorem).

Idempotents and decompositions

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To give a direct sum decomposition of a module into submodules is the same as to give orthogonal idempotents in the endomorphism ring of the module that sum up to the identity map.[1] Indeed, if  , then, for each  , the linear endomorphism   given by the natural projection followed by the natural inclusion is an idempotent. They are clearly orthogonal to each other (  for  ) and they sum up to the identity map:

 

as endomorphisms (here the summation is well-defined since it is a finite sum at each element of the module). Conversely, each set of orthogonal idempotents   such that only finitely many   are nonzero for each   and   determine a direct sum decomposition by taking   to be the images of  .

This fact already puts some constraints on a possible decomposition of a ring: given a ring  , suppose there is a decomposition

 

of   as a left module over itself, where   are left submodules; i.e., left ideals. Each endomorphism   can be identified with a right multiplication by an element of R; thus,   where   are idempotents of  .[2] The summation of idempotent endomorphisms corresponds to the decomposition of the unity of R:  , which is necessarily a finite sum; in particular,   must be a finite set.

For example, take  , the ring of n-by-n matrices over a division ring D. Then   is the direct sum of n copies of  , the columns; each column is a simple left R-submodule or, in other words, a minimal left ideal.[3]

Let R be a ring. Suppose there is a (necessarily finite) decomposition of it as a left module over itself

 

into two-sided ideals   of R. As above,   for some orthogonal idempotents   such that  . Since   is an ideal,   and so   for  . Then, for each i,

 

That is, the   are in the center; i.e., they are central idempotents.[4] Clearly, the argument can be reversed and so there is a one-to-one correspondence between the direct sum decomposition into ideals and the orthogonal central idempotents summing up to the unity 1. Also, each   itself is a ring on its own right, the unity given by  , and, as a ring, R is the product ring  

For example, again take  . This ring is a simple ring; in particular, it has no nontrivial decomposition into two-sided ideals.

Types of decomposition

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There are several types of direct sum decompositions that have been studied:

  • Semisimple decomposition: a direct sum of simple modules.
  • Indecomposable decomposition: a direct sum of indecomposable modules.
  • A decomposition with local endomorphism rings[5] (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are local rings (a ring is local if for each element x, either x or 1 − x is a unit).
  • Serial decomposition: a direct sum of uniserial modules (a module is uniserial if the lattice of submodules is a finite chain[6]).

Since a simple module is indecomposable, a semisimple decomposition is an indecomposable decomposition (but not conversely). If the endomorphism ring of a module is local, then, in particular, it cannot have a nontrivial idempotent: the module is indecomposable. Thus, a decomposition with local endomorphism rings is an indecomposable decomposition.

A direct summand is said to be maximal if it admits an indecomposable complement. A decomposition   is said to complement maximal direct summands if for each maximal direct summand L of M, there exists a subset   such that

 [7]

Two decompositions   are said to be equivalent if there is a bijection   such that for each  ,  .[7] If a module admits an indecomposable decomposition complementing maximal direct summands, then any two indecomposable decompositions of the module are equivalent.[8]

Azumaya's theorem

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In the simplest form, Azumaya's theorem states:[9] given a decomposition   such that the endomorphism ring of each   is local (so the decomposition is indecomposable), each indecomposable decomposition of M is equivalent to this given decomposition. The more precise version of the theorem states:[10] still given such a decomposition, if  , then

  1. if nonzero, N contains an indecomposable direct summand,
  2. if   is indecomposable, the endomorphism ring of it is local[11] and   is complemented by the given decomposition:
      and so   for some  ,
  3. for each  , there exist direct summands   of   and   of   such that  .

The endomorphism ring of an indecomposable module of finite length is local (e.g., by Fitting's lemma) and thus Azumaya's theorem applies to the setup of the Krull–Schmidt theorem. Indeed, if M is a module of finite length, then, by induction on length, it has a finite indecomposable decomposition  , which is a decomposition with local endomorphism rings. Now, suppose we are given an indecomposable decomposition  . Then it must be equivalent to the first one: so   and   for some permutation   of  . More precisely, since   is indecomposable,   for some  . Then, since   is indecomposable,   and so on; i.e., complements to each sum   can be taken to be direct sums of some  's.

Another application is the following statement (which is a key step in the proof of Kaplansky's theorem on projective modules):

  • Given an element  , there exist a direct summand   of   and a subset   such that   and  .

To see this, choose a finite set   such that  . Then, writing  , by Azumaya's theorem,   with some direct summands   of   and then, by modular law,   with  . Then, since   is a direct summand of  , we can write   and then  , which implies, since F is finite, that   for some J by a repeated application of Azumaya's theorem.

In the setup of Azumaya's theorem, if, in addition, each   is countably generated, then there is the following refinement (due originally to Crawley–Jónsson and later to Warfield):   is isomorphic to   for some subset  .[12] (In a sense, this is an extension of Kaplansky's theorem and is proved by the two lemmas used in the proof of the theorem.) According to (Facchini 1998), it is not known whether the assumption "  countably generated" can be dropped; i.e., this refined version is true in general.

Decomposition of a ring

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On the decomposition of a ring, the most basic but still important observation, known as the Wedderburn-Artin theorem is this: given a ring R, the following are equivalent:

  1. R is a semisimple ring; i.e.,   is a semisimple left module.
  2.   for division rings  , where   denotes the ring of n-by-n matrices with entries in  , and the positive integers  , the division rings  , and the positive integers   are determined (the latter two up to permutation) by R
  3. Every left module over R is semisimple.

To show 1.   2., first note that if   is semisimple then we have an isomorphism of left  -modules   where   are mutually non-isomorphic minimal left ideals. Then, with the view that endomorphisms act from the right,

 

where each   can be viewed as the matrix ring over  , which is a division ring by Schur's Lemma. The converse holds because the decomposition of 2. is equivalent to a decomposition into minimal left ideals = simple left submodules. The equivalence 1.   3. holds because every module is a quotient of a free module, and a quotient of a semisimple module is semisimple.

See also

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Notes

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  1. ^ Anderson & Fuller 1992, Corollary 6.19. and Corollary 6.20.
  2. ^ Here, the endomorphism ring is thought of as acting from the right; if it acts from the left, this identification is for the opposite ring of R.
  3. ^ Procesi 2007, Ch.6., § 1.3.
  4. ^ Anderson & Fuller 1992, Proposion 7.6.
  5. ^ (Jacobson 2009, A paragraph before Theorem 3.6.) calls a module strongly indecomposable if nonzero and has local endomorphism ring.
  6. ^ Anderson & Fuller 1992, § 32.
  7. ^ a b Anderson & Fuller 1992, § 12.
  8. ^ Anderson & Fuller 1992, Theorrm 12.4.
  9. ^ Facchini 1998, Theorem 2.12.
  10. ^ Anderson & Fuller 1992, Theorem 12.6. and Lemma 26.4.
  11. ^ Facchini 1998, Lemma 2.11.
  12. ^ Facchini 1998, Corollary 2.55.

References

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  • Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487
  • Frank W. Anderson, Lectures on Non-Commutative Rings Archived 2021-06-13 at the Wayback Machine, University of Oregon, Fall, 2002.
  • Facchini, Alberto (16 June 1998). Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules. Springer Science & Business Media. ISBN 978-3-7643-5908-9.
  • Jacobson, Nathan (2009), Basic algebra, vol. 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7
  • Y. Lam, Bass's work in ring theory and projective modules [MR 1732042]
  • Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 9780387260402.
  • R. Warfield: Exchange rings and decompositions of modules, Math. Annalen 199(1972), 31–36.