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)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)