Abstract
Let R be a commutative ring. We investigate R-modules which can be written as finite sums of second R-submodules (we call them second representable). The class of second representable modules lies between the class of finitely generated semisimple modules and the class of representable modules; moreover, we give examples to show that these inclusions are strict even for Abelian groups. We provide sufficient conditions for an R-module M to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (main) second attached prime ideals related to a module with such a presentation.
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