On Some Generalizations of Reversible and Semicommutative Rings

Year 2017, , 11 - 27, 11.07.2017

Arnab Bhattacharjee Uday Shankar Chakraborty

https://doi.org/10.24330/ieja.325916

Abstract

The concept of strongly central reversible rings has been
introduced in this paper. It has been shown that the class of
strongly central reversible rings properly contains the class of
strongly reversible  rings and is properly contained in the class
of central reversible rings. Various properties of the
above-mentioned rings have been investigated. The concept of
strongly central semicommutative rings has also been introduced
and its relationships with other rings have been studied. Finally
an open question raised in [D. W. Jung, N. K. Kim, Y. Lee and S.
J. Ryu, Bull. Korean Math. Soc., 52(1) (2015), 247-261] has been
answered.

Keywords

Strongly central reversible ring, strongly central semicommutative ring

References

• N. Agayev, A. Harmanci and S. Halicioglu, Extended Armendariz rings, Alge- bras Groups Geom., 26(4) (2009), 343-354.
• N. Agayev, G. Gungoroglu, A. Harmanci and S. Halicioglu, Central Armen- dariz rings, Bull. Malays. Math. Sci. Soc., 34(1) (2011), 137-145.
• D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra, 26(7) (1998), 2265-2272.
• R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra, 319(8) (2008), 3128-3140.
• H. J. Cha, D. W. Jung, H. K. Kim, J. A. Kim, C. I. Lee, Y. Lee, S. B. Nam, S. J. Ryu, Y. Seo, H. J. Sung and S. J. Yun, On a ring property generalizing power- Armendariz and central Armendariz rings, Korean J. Math., 23(3) (2015), 337-355.
• W. Chen, On nil-semicommutative rings, Thai J. Math., 9(1) (2011), 39-47.
• P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(6) (1999), 641-648.
• C. Huh, Y. Lee and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra, 30(2) (2002), 751-761.
• D. W. Jung, N. K. Kim, Y. Lee and S. J. Ryu, On properties related to re- versible rings, Bull. Korean Math. Soc., 52(1) (2015), 247-261.
• G. Kafkas, B. Ungor, S. Halicioglu and A. Harmanci, Generalized symmetric rings, Algebra Discrete Math., 12(2) (2011), 72-84.
• N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra, 223(2) (2000), 477-488.
• N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra, 185(1-3) (2003), 207-223.
• N. K. Kim, T. K. Kwak and Y. Lee, Semicommutative property on nilpotent products, J. Korean Math. Soc., 51(6) (2014), 1251-1267.
• H. Kose, B. Ungor, S. Halicioglu and A. Harmanci, A generalization of re- versible rings, Iran. J. Sci. Technol. Trans. A Sci., 38(1) (2014), 43-48.
• J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull., 14 (1971), 359-368.
• L. Liang, L. Wang and Z. Liu, On a generalization of semicommutative rings, Taiwanese J. Math., 11(5) (2007), 1359-1368.
• L. Motais de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), Bib. Nat., Paris (1982), 71-73.
• P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra, 298(1) (2006), 134-141.
• L. Ouyang and H. Chen, On weak symmetric rings, Comm. Algebra, 38(2) (2010), 697-713.
• T. Ozen, N. Agayev and A. Harmanci, On a class of semicommutative rings, Kyungpook Math. J., 51(3) (2011), 283-291.
• Y. Qu and J.Wei, Some notes on nil-semicommutative rings, Turkish J. Math., 38(2) (2014), 212-224.
• M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci., 73(1) (1997), 14-17.
• B. Ungor, S. Halicioglu, H. Kose and A. Harmanci, Rings in which every nilpotent is central, Algebras Groups Geom., 30(1) (2013), 1-18.
• L. Wang and J. C. Wei, Central semicommutative rings, Indian J. Pure Appl. Math., 45(1) (2014), 13-25.
• G. Yang and R. Du, Rings over which polynomial rings are semi-commutative, Vietnam J. Math., 37(4) (2009), 527-535.
• G. Yang and Z. K. Liu, On strongly reversible rings, Taiwanese J. Math., 12(1) (2008), 129-136.