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Abstract: A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, “a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an interesting natural question of this sort is “whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?” The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. [Copyright &y& Elsevier]

Let (R,m) be a Noetherian local ring of prime characteristic p. In this paper, as an extension of the concept of pn-basis, we introduce the notion of m-adic pn-basis, and we show that R/Rpn has an m-adic pn-basis for every n (n=1,2,…) if and only if R is a regular local ring. For any Noetherian local ring, further we introduce the concept of m-adic free graded algebra and we give a regularity criterion in terms of the higher differential algebra, and others. [Copyright &y& Elsevier]