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REGULARITY OF THE GENERALIZED CENTROID OF SEMI-PRIME GAMMA RINGS

  • Ali Ozturk, Mehmet (Department of Mathematics Faculty of Arts and Science Cumhuriyet University) ;
  • Jun, Young-Bae (Department of Mathematics Education and RINS Gyeongsang National University)
  • Published : 2004.04.01

Abstract

The aim of this note is to study properties of the generalized centroid of the semi-prime gamma rings. Main results are the following theorems: (1) Let M be a semi-prime $\Gamma$-ring and Q a quotient $\Gamma$-ring of M. If W is a non-zero submodule of the right (left) M-module Q, then $W\Gamma$W $\neq 0. Furthermore Q is a semi-prime $\Gamma$-ring. (2) Let M be a semi-prime $\Gamma$-ring and $C_{{Gamma}$ the generalized centroid of M. Then $C_{\Gamma}$ is a regular $\Gamma$-ring. (3) Let M be a semi-prime $\Gamma$-ring and $C_{\gamma}$ the extended centroid of M. If $C_{\gamma}$ is a $\Gamma$-field, then the $\Gamma$-ring M is a prime $\Gamma$-ring.

Keywords

semi-prime $\Gamma$-ring;generalized centroid;(regular;prime;quotient) $\Gamma$-ring.

References

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Cited by

  1. The Extended Centroid of the Prime Gamma Semirings vol.0, pp.0, 2014, https://doi.org/10.2478/aicu-2014-0034