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Une caracterisation des anneaux artiniens √† ideaux principaux. (A characterization of Artinian principal ideal rings). (French) Zbl 0655.13013

Ring theory, Proc. Conf., Granada/Spain 1986, Lect. Notes Math. 1328, 245-254 (1988).
[For the entire collection see Zbl 0638.00012.]
It is well-known that injective endomorphisms of artinian modules are also surjective and that surjective endomorphisms of noetherian modules are injective. The subject of the present paper is to specify those commutative rings over which a module is artinian (noetherian) in case each injective (surjective) endomorphism is an automorphism. As a main result it is shown that each of these properties characterizes artinian principal ideal rings. The proof consists in the construction of a suitable module over an artinian non-principal ideal ring as well as in an application of a theorem by I. S. Cohen and I. Kaplansky [Math. Z. 54, 97-101 (1951; Zbl 0043.267)].
Reviewer: W.Zimmermann

MSC:

13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13F10 Principal ideal rings
13E05 Commutative Noetherian rings and modules