Let be a commutative ring with 1 and let be a unital left module. Prove that is irreducible if and only if for some maximal ideal . (Where means “isomorphic as an -module.”)
Suppose is irreducible, fix nonzero, and define by . Note that for all and , . So is an -module homomorphism. Now recall that since and is irreducible, then by this previous exercise, . In particular, if , then there exists such that . Thus is surjective.
FInally, we claim that is a maximal ideal. To that end, let be nonzero. In particular, . Since is irreducible, we have . In particular, for some . Then , so that . That is, . So every nonzero element of has a left inverse. Since is commutative, is a field, so that is a maximal ideal of .
By the First Isomorphism Theorem, we have , where is a maximal ideal.
Conversely, suppose is a maximal ideal. Now is a field. Let be nonzero. Then there exists such that . Now let . Note that , so that . That is, is generated (as an -module) by any nonzero element. By this previous exercise, is an irreducible -module.