Homogeneous idealization and some dual notions around comultiplication modules

Abstract: Let R be a commutative ring with identity, and let M be a unital R-module. D.D.Anderson proved that a submodule N of M is multiplication if and only if $0_{(+)}N$ is a multiplication ideal of $R_{(+)}M$, the homogeneous idealization of M. In this article, we show that a similar statement holds for comultiplication modules. We develop the tool of idealization of a module particularly in the context of cocyclic modules, self-cogenerator modules, comultiplication modules (self-cogenerated modules), couniform modules, $AB5^*$ modules, direct family and inverse family of submodules.

Keywords: Cocyclic module, Self-cogenerator module, Comultiplication module, Nonsingular module

Classification (MSC2000): 13C13; 13C99

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