Semisimple module

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In abstract algebra, a module is said to be semisimple if it is the sum of simple submodules. Here, the base ring is a ring with unity, though not necessarily commutative. A semisimple module may be characterised as being (isomorphic to) a direct sum of simple modules.

1 Properties
2 Semisimple Rings
3 Semisimple vs Simple Rings
4 Examples
5 See also
6 Further reading

[edit] Properties

Every submodule of a semisimple module is a direct summand.

If M is semisimple and N is a submodule, then N and M/N are also semisimple.

If each $M i$ is a semisimple module, then so is $\oplus_i M_i$ .

A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero.

[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.

[edit] Examples

If k is a field and G is a finite group of order n, then the group ring $k [G]$ is semisimple if and only if the characteristic of k does not divide n. This is an important result in group representation theory.

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 $D i$ is a division ring and $M n (D)$ is the ring of n-by-n matrices with entries in D.