Une caracterisation des anneaux artiniens a ideaux principaux

Abstract

Let be A a ring and M an A-Module. We say that M satisfies the property(I) (resp. (S)) if every injective (resp. surjective) A-endomorphism of M is an automorphism. It is well known that every Artinian (resp. Noetherian) module satisfies the property (I) (rep. (S)). The converse is not true (for example the Z-module Q of rational numbers has the properties (I) and (S), but Q is neither Artinian nor Noetherian, regarded as Z-module). The main aim of this paper is to give a characterization of commutative rings A with the property that every A-module satisfying (I) (resp. (S)) is Artinian (resp. Noetherian). We first show that if A is a non principal Artinian commutative ring, then there exists a non finitely generated A-module whose endomorphism ring E is local and J² = 0, where J is the Jacobson radical of E (prop. 7). This result enables us to show that, for a commutative ring A, the following conditions are equivalent:

• a) Every A-module satisfying the property (I) is Artinian.

• b) Avery A-module satisfying the property (S) is Northerian.

• c) A is an Artinian principal ideal ring (th. 9).

Finally we show, by an example that the result above falls in general if A is not commutative.