Commutative Local Rings Whose Ideals Are Direct Sums of Cyclics

Details Commutative Local Rings Whose Ideals Are Direct Sums of Cyclics Commutative Local Rings Whose Ideals Are Direct Sums of Cyclics

2012-01-29

arxiv.org/abs/1201.6076v1

Mathematics

Commutative Algebra

14 pages

Scientific paper

A well-known result of Kothe 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". This motivated us to ask the following question: "whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?" More recently, this question was answered by Behboodi et al., in [J. Algebra 345 (2011) 257-265] for the case R is a finite direct product of commutative Noetherian local rings. The goal of this paper is to answer this question in the case R is a finite direct product of commutative local rings (not necessarily Noetherian) and the structure of such rings is completely described.

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