Some Remarks on Primal Submodules
DOI:
https://doi.org/10.5644/SJM.04.2.03Keywords:
Primal submodule, primary submoduleAbstract
In this paper, we study the primal submodules of a module over a commutative ring with non-zero identity. We generalize the primal decomposition of ideals (see [2]) to that of submodules. Let $R$ be a commutative ring, $M$ an $R$-module and $N$ a submodule of $M$. We establish a decomposition of $N$ as an intersection of primal submodules of $M$. We show that if $R$ is a Prüfer domain of finite character, then $N$ has a primal decomposition. Also we prove that the representation of submodules as reduced intersections of primal submodules is unique.
2000 Mathematics Subject Classification. 13A05, 13F05, 20M14
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References
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L. Fuchs and E. Mosteig, Ideal theory in Pr¨ufer domains-An unconventional approach, J. Algebra, 252 (2002), 411–430.
C. U. Jensen, Arithmetical rings, Acta Math. Hung., 17 (1-2) (1966), 115–123.
R. Y. Sharp, Steps in Commutative Algebra, Cambridge University Press, Cambridge, 1990.
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