ON JOINTLY PRIME RADICALS OF (R,S)-MODULES

Dian Ariesta Yuwaningsih, Indah Emilia Wijayanti

Abstract


Let $M$ be an $(R,S)$-module. In this paper a generalization of the m-system set of modules to $(R,S)$-modules is given. Then for an $(R,S)$-submodule $N$ of $M$, we define $\sqrt[(R,S)]{N}$ as the set of $a\in M$ such that every m-system containing $a$ meets $N$. It is shown that $\sqrt[(R,S)]{N}$ is the intersection of all jointly prime $(R,S)$-submodules of $M$ containing $N$. We define jointly prime radicals of an $(R,S)$-module $M$ as $rad_{(R,S)}(M)=\sqrt[(R,S)]{0}$. Then we present some properties of jointly prime radicals of an $(R,S)$-module.

DOI : http://dx.doi.org/10.22342/jims.21.1.199.25-34


Keywords


(R, S)-module, jointly prime (R, S)-submodule, m-system, prime radical.

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References


Behboodi, M., "On the Prime Radical and Baer's Lower Nilradical of Modules", Acta Mathematica Hungarica, 122 (2009), 293-306.

Goodearl, K., R. and Wareld, R.B., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2004.

Khumprapussorn, T., Pianskool, S., and Hall, M., (R,S)-Modules and their Fully and Jointly Prime Submodules, International Mathematical Forum, 7 (2012), 1631-1643.

Lam, T.Y., A First Course in Noncommutative Rings, Springer-Verlag New York, Inc., 2001.




DOI: https://doi.org/10.22342/jims.21.1.199.25-34

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