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2010•Maǧallaẗ al-rāfidayn li-ʿulūm al-ḥāsibāt wa-al-riyāḍiyyāẗ/˜Al-œRafidain journal for computer sciences and mathematicsOpen access

On Rings Whose Simple Singular R-Modules Are Flat, I

Raida Mahmood, Abdullah M. Abdul-Jabbar

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Abstract

In this paper we investigate von Neumann regularity of rings whose simple singular right R-modules are flat. It is proved that a ring R is strongly regular if and only if R is a semiprime right quasi-duo ring whose simple singular right R-modules are flat. Moreover, it is shown that if R is a P. I.-ring whose simple singular right R-modules are flat, then R is a strongly p-regular ring. Finally, it is shown that if R is a right continuous ring whose simple singular right R-modules are flat, then R is a von Neumann regular ring.

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In this paper we investigate von Neumann regularity of rings whose simple singular right R-modules are flat. It is proved that a ring R is strongly regular if and only if R is a semiprime right quasi-duo ring whose simple singular right R-modules are flat. Moreover, it is shown that if R is a P. I.-ring whose simple singular right R-modules are flat, then R is a strongly p-regular ring. Finally, it is shown that if R is a right continuous ring whose simple singular right R-modules are flat, then R is a von Neumann regular ring.

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Available abstract

In this paper we investigate von Neumann regularity of rings whose simple singular right R-modules are flat. It is proved that a ring R is strongly regular if and only if R is a semiprime right quasi-duo ring whose simple singular right R-modules are flat. Moreover, it is shown that if R is a P. I.-ring whose simple singular right R-modules are flat, then R is a strongly p-regular ring. Finally, it is shown that if R is a right continuous ring whose simple singular right R-modules are flat, then R is a von Neumann regular ring.

Key concepts: Mathematics, Simple (philosophy), Simple ring, Von Neumann regular ring, Ring (chemistry), Principal ideal ring, Regular ring, Primitive ring

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