ON LEFT PRIMARY AND WEAKLY LEFT PRIMARY IDEALS IN $\GAMMA$-LA- RINGS

Abstract

In this paper, we study left ideals, left primary and weakly left primary ideals in $\Gamma$-LA-rings. Some characterizations of left primary and weakly left primary ideals are obtained. Moreover, we investigate relationships left primary and weakly left primary ideals in $\Gamma$-LA-rings. Finally, we obtain necessary and sufficient conditions of a weakly left primary ideal to be a left primary ideals in $\Gamma$-LA- rings.

Keywords

• $\Gamma$-LA-ring,left primary ideal,weakly left primary ideal,left ideal

LA- Rings

Abstract

References

1. M.A. Kazim and M. Naseeruddin, On almost semigroups, The Alig. Bull. Math., 2(1972), 1-7.
2. M. Khan, V. Amjid and Faisal, Characterizations of intra-regular -AG**-groupoids by the properties of their -ideals, arXiv:1011.1364v1 [math.GR], (2010).
3. M. Khan and Naveed Ahmad, Characterizations of left almost semigroups by their ideals, Journal of Advanced Research in Pure Mathematics, 2(3)(2010), 61-73.
4. Q. Mushtaq and S.M. Yousuf, On LA-semigroups, The Alig. Bull. Math., 8(1978), 65-70.
5. Q. Mushtaq and S.M. Yousuf, On LA-semigroup defined by a commutative inverse semigroup, Math. Bech., 40(1988), 59-62. [6] M. Naseeruddin, Some semigroups and flocks, Ph.D. thesis Aligarh Muslim University Aligarh India, (1970). almost
6. P.V. Protic and N. Stevanovic, AG-test and some general properties of Abel-Grassmann's groupoids, PU. M. A., 4(6)(1995), 371-383.
7. I. Rehman, M. Shah, T. Shah and A. Razzaque, On existence of nonassociative LA-ring, Analele Stiintfice ale Universitatii Ovidius Constanta, 21(3)(2013), 223-228.
8. M.K. Sen, On -semigroups, Proceeding of International Symposium on Algebra and Its Applications Decker Publication New York, (1981), 301-308.

9. M.K. Sen and N.K. Saha, On -semigroups I, Bull. Cal. Math. Soc., 78(1986), 180-186.
10. T. Shah, G. Ali and Fazal ur Rehman, Direct sum of ideals in a generalized LA-ring, International Mathematical Forum, 6(22) (2011), 1095-1101.
11. T. Shah, Fazal ur Rehman and M. Raees, On Near Left Almost Rings, (to appear).
12. T. Shah, N. Kausar and I. Rehman, fuzzy normal subrings over a nonassociative ring, Analele Stiintfice ale Universitatii Ovidius Constanta, 20(1)(2012), 369–386.
13. T. Shah and I. Rehman, On -Ideals and -Bi- Ideals in -AG-groupoids, International Journal of Algebra, 4(6)(2010), 267-276.
14. T. Shah and I. Rehman, On LA-rings of finitely nonzero functions, Int. J. Contemp. Math. Sciences, 5(5) (2010), 209-222.
15. M. Shah and T. Shah, Some basic properties of LA- ring, 6(44)(2011), 2195-2199. Forum,