Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Extensions of Strongly α-semicommutative Rings

  • Ayoub, Elshokry (Department of Mathematics, Northwest Normal University) ;
  • Ali, Eltiyeb (Department of Mathematics, Faculty of Education, University of Khartoum) ;
  • Liu, ZhongKui (Department of Mathematics, Northwest Normal University)
  • Received : 2017.08.30
  • Accepted : 2018.03.09
  • Published : 2018.06.23
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Abstract

This paper is devoted to the study of strongly ${\alpha}-semicommutative$ rings, a generalization of strongly semicommutative and ${\alpha}-rigid$ rings. Although the n-by-n upper triangular matrix ring over any ring with identity is not strongly ${\bar{\alpha}}-semicommutative$ for $n{\geq}2$, we show that a special subring of the upper triangular matrix ring over a reduced ring is strongly ${\bar{\alpha}}-semicommutative$ under some additional conditions. Moreover, it is shown that if R is strongly ${\alpha}-semicommutative$ with ${\alpha}(1)=1$ and S is a domain, then the Dorroh extension D of R by S is strongly ${\bar{\alpha}}-semicommutative$.

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References

  1. M. Baser, A. Harmanci, T. K. Kwak, Generalized semicommutative rings and their extensions, Bull. Korean Math. Soc., 45(2)(2008), 285-297. https://doi.org/10.4134/BKMS.2008.45.2.285
  2. M. Baser, C. Y. Hong and T. K. Kwak, On extended reversible rings, Algebra Colloq., 16(1)(2009), 37-48. https://doi.org/10.1142/S1005386709000054
  3. M. Baser and T. K. Kwak, Extended semicommutative rings, Algebra Colloq., 17(2)(2010), 257-264. https://doi.org/10.1142/S1005386710000271
  4. P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(1999), 641-648. https://doi.org/10.1112/S0024609399006116
  5. Y. Gang and D. Juan, Rings over which polynomial rings are semicommutative, Vietnam J. Math., 37(4)(2009), 527-535.
  6. C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra, 151(3)(2000), 215-226. https://doi.org/10.1016/S0022-4049(99)00020-1
  7. C. Y. Hong, T.K. Kwak, S. T. Rizvi, Extensions of generalized Armendariz rings, Algebra Colloq., 13(2)(2006) 253-266. https://doi.org/10.1142/S100538670600023X
  8. C. Huh, Y. Lee, A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra, 30(2)(2002), 751-761. https://doi.org/10.1081/AGB-120013179
  9. D. A. Jordan, Bijective extensions of injective ring endomorphisms, J. London Math. Soc., 35(2)(1982), 435-448.
  10. N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra, 185(2003), 207-223. https://doi.org/10.1016/S0022-4049(03)00109-9
  11. J. Krempa, Some examples of reduced rings, Algebra Colloq., 3(4)(1996), 289-300.
  12. T. K. Kwak, Y. Lee and S. J. Yun, The Armendariz property on ideals, J. Algebra, 354(2012), 121-135. https://doi.org/10.1016/j.jalgebra.2011.12.019
  13. Z. K. Liu, Semicommutative Subrings of Matrix Rings, J. Math. Res. Exposition, 26(2)(2006), 264-268.
  14. Z. K. Liu and R. Y. Zhao, On Weak Armendariz Rings, Comm. Algebra, 34(2006), 2607-2616. https://doi.org/10.1080/00927870600651398
  15. P. Patricio, R. Puystjens, About the von Neumann regularity of triangular block matrices, Linear Algebra Appl., 332/334(2001), 485-502. https://doi.org/10.1016/S0024-3795(01)00295-6
  16. H. Pourtaherian and I. S. Rakhimov, On skew version of reversible rings, Int. J. Pure Appl. Math., 73(3)(2011), 267-280.
  17. M. B. Rege, S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci., 73(1997), 14-17. https://doi.org/10.3792/pjaa.73.14
  18. G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc., 184(1973), 43-60. https://doi.org/10.1090/S0002-9947-1973-0338058-9
  19. G. Yang, Semicommutative and reduced rings, Vietnam J. Math.. 35(3)(2007), 309-315.
  20. G. Yang, Z. K. Liu, On strongly reversible rings, Taiwanese J. Math., 12(1)(2008), 129-136. https://doi.org/10.11650/twjm/1500602492