Toma Albu and Patrick F. Smith: Primal, completely irreducible, and primary meet decompositions in modules. p.297-311

Abstract:

This paper was inspired by the work of Fuchs, Heinzer, and Olberding concerning primal and completely irreducible ideals. It is proved that if $R$ is a commutative Noetherian ring then every primal submodule of an $R$-module $M$ is a primary submodule of $M$ if and only if for all prime ideals $\,\mathfrak{p}\subset\mathfrak{q}\,$ of $R$, every $\,\mathfrak{p}$-primary submodule of $M$ is contained in every $\,\mathfrak{q}$-primary submodule of $M$. Moreover, for a commutative Noetherian ring $R$, every primal ideal of $R$ is primary if and only if $R$ is a finite direct product of Artinian rings and one-dimensional domains. Given a general ring $R$, a right $R$-module $M$ has the property that every submodule contains a completely coirreducible submodule if and only if the Jacobson radical of any non-zero submodule $N$ of $M$ is zero and an irredundant intersection of maximal submodules of $N$. The paper closes with seven open problems.

Key Words: Primal submodule, irreducible submodule, completely irreducible submodule, completely coirreducible module, primary submodule, primary decomposition, completely irreducible decomposition, Noetherian ring.

2000 Mathematics Subject Classification: Primary: 13C13;
Secondary: 13C99, 13E05, 16D80.

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