Definition
A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
For a module M, the following are equivalent:
- M is a direct sum of irreducible modules.
- M is the sum of its irreducible submodules.
- 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, the starting idea is to find an irreducible submodule by picking any and letting be a maximal submodule such that . It can be shown that the complement of is irreducible.
Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.
Read more about this topic: Semisimple Module
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)