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| Paper IPM / M / 17655 |
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| Abstract: | |||||
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For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal M of R is of the form M =Rx â??Ry â??(óâ?? Rsó), where R/Ann(x) and R/Ann(y) are principal ideal rings and s2 ó = 0 for all óâ??s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules.
We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal M in the above-mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.
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