Semisimple module

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[edit] Definition

In abstract algebra, a module is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.

This is stronger than completely decomposable, which is a direct sum of indecomposable submodules).

Here, the base ring is a ring with unity, though not necessarily commutative.

[edit] Equivalent definitions

For a module M, the following are equivalent:

  1. M is a direct sum of irreducible modules.
  2. M is the sum of its irreducible submodules.
  3. Every submodule of M is a direct summand: for every submodule N of M, there is a complement P such that M = N ⊕ P.

For 3 \implies 2, the starting idea is to find an irreducible submodule by picking any x\in M and letting P be a maximal submodule such that x \notin P. It can be shown that the complement of P is irreducible.[1]

[edit] Properties

  • If M is semisimple and N is a submodule, then N and M/N are also semisimple.
  • If each Mi is a semisimple module, then so is \bigoplus_i M_i.

[edit] Semisimple rings

A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. Hence, one often drops the left/right quantifier altogether and simply speaks of semisimple rings.

Semisimple rings are of particular interest to algebraists. For example, if the base ring R is semisimple, then all R-modules would automatically be semisimple. Furthermore, every simple (left) R-module is isomorphic to a minimal left ideal of R.

Semisimple rings are also small (they're both Artinian and Noetherian). From the above properties, a ring is semisimple if and only if it is Artinian and its radical is zero.

[edit] Semisimple vs simple rings

One should beware that despite the terminology, not all simple rings are semisimple. The problem is that the ring may be "too big", and possibly not (left/right) Artinian. In fact, if R is a simple ring with a minimal left/right ideal, then R is semisimple.

Classic examples of simple, but not semisimple, rings are the Weyl algebras, such as Q<x,y>/(xy-yx-1) which is a simple noncommutative domain. These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings. The module theory for the Weyl algebras is well studied and differs significantly from that of semisimple rings.

[edit] Examples

  • By the Artin-Wedderburn_theorem, a ring R is semisimple if and only if it is (isomorphic to) M_n(D_1) \times M_n(D_2) \times \dots \times M_n(D_r), where each Di is a division ring and Mn(D) is the ring of n-by-n matrices with entries in D.

[edit] See also

[edit] References

  1. ^ Nathan Jacobson, Basic Algebra II (Second Edition), p.120

[edit] Further reading

  • R.S. Pierce. Associative Algebras. Graduate Texts in Mathematics vol 88.
  • T.Y. Lam. A First Course in Non-commutative Rings. Graduate Texts in Mathematics vol 131.