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ABSTRACT
In this paper, unless otherwise stated, all rings are commutative with identity and all modules are unital. We give sufficient conditions to ensure that a submodule has a module-reduced primary decomposition. In general, the radical of a primary submodule is not prime and the radical does not split intersections of submodules, as is valid in the ideal case. We study sufficient conditions for which these properties hold in the module setting. These conditions involve dimension arguments, consideration of finitely generated modules, and the spectrum of a given prime ideal. Further, we consider the computational problem of finding a Gröbner basis of both the colon and the radical of a submodule. A characterization of the elements of the colon is given, along with a method of computing the radical of a submodule in certain cases.
ACKNOWLEDGMENT
The authors are grateful for the referee's valuable comments and suggestions concerning their oversights.