On some classes of rings related to the Fitting lemma (Q1199651)

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.