## On some classes of rings related to the Fitting lemma. (Sur quelques classes d’anneaux lieés au lemme de Fitting.)(French)Zbl 0816.16022

Let $$R$$ be a ring and $$M$$ a left $$R$$-module. $$M$$ is said to satisfy the condition (I) (resp. the condition (S)) if every monomorphic (resp. epimorphic) endomorphism $$f$$ of $$M$$ is an automorphism. $$M$$ is said to satisfy the condition (F) if for any endomorphism $$f$$ of $$M$$, we have $$M = \text{Im }f^ n \oplus \text{Ker }f^ n$$ for some integer $$n \geq 1$$. This paper studies three classes of rings $$R$$ satisfying the following conditions, respectively: (a) Every left $$R$$-module satisfying (I) is Artinian; (b) Every left $$R$$-module satisfying (S) is Noetherian; (c) Every left $$R$$-module satisfying (F) is of finite length. It is shown that if $$R$$ is a commutative ring, then $$R$$ satisfies either of the above three conditions if and only if $$R$$ is an Artinian principal ideal ring. Certain group rings which satisfy these conditions are also studied.

### MSC:

 16P20 Artinian rings and modules (associative rings and algebras) 16P40 Noetherian rings and modules (associative rings and algebras) 16W20 Automorphisms and endomorphisms 13E10 Commutative Artinian rings and modules, finite-dimensional algebras
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### References:

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