A new characterization of commutative Artinian rings (Q1775327)

Let \(R\) be a commutative Noetherian ring and \(M\) a \(R\)-module. \(M\) is good if \((0)\) has a primary decomposition in \(M\). \(M\) is called secondary if for any \(x\in R\) the multiplication \(M\rightarrow M\) by \(x\) is either surjective or nilpotent. \(M\) is called representable if it is a sum of secondary submodules. The aim of this paper is to show that the following statements are equivalent: (i) \(R\) is Artinian ring, (ii) Every \(R\)-module is good, (iii) Every \(R\)-module is representable, (iv) Every nonzero submodule of any representable \(R\) module is representable. In particular this answers positively a question of \textit{S. Ebrahimi-Atami} [Int. J. Math. Math. Sci. 31, 321--327 (2002; Zbl 1008.13002)].