Ring endomorphisms with nil-shifting property
الموضوعات :C. A. K. Ahmed 1 , R. T. M. Salim 2
1 - Department of Mathematics, University of Zakho, Kurdistan Region, Iraq
2 - Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq
الکلمات المفتاحية: CNZ ring&lrm, , &lrm, reversible ring&lrm, , &lrm, matrix ring&lrm, , &lrm, polynomial ring,
ملخص المقالة :
Cohn called a ring $R$ is reversible if whenever $ab = 0,$ then $ba = 0$ for $a,b\in R.$ The reversible property is an important role in noncommutative ring theory. Recently, Abdul-Jabbar et al. studied the reversible ring property on nilpotent elements, introducingthe concept of commutativity of nilpotent elements at zero (simply, a CNZ ring). In this paper, we extend the CNZ property of a ring as follows: Let $R$ be a ring and $\alpha$ an endomorphism of $R$, we say that $ R $ is right (resp., left) $\alpha$-nil-shifting ring if whenever $ a\alpha(b) = 0 $ (resp., $\alpha(a)b = 0$) for nilpotents $a,b$ in $R$, $ b\alpha(a) = 0 $ (resp., $ \alpha(b)a= 0) $. The characterization of $\alpha$-nil-shifting rings and their related properties are investigated.
[1] A. M. Abdul-Jabbar, C. A. K. Ahmed, T. K. Kwak, Y. Lee, On Commutativity of nilpotent elements at zero, Comm. Korean. Math. Soc. 32 (4) (2017), 811-826.
[2] A. M. Abdul-Jabbar, C. A. K. Ahmed, T. K. Kwak, Y. Lee, Y. J. Seo, Zero commutativity of nilpotent elements skewed by ring endomorphisms, Comm. Algebra. 45 (11) (2017), 4881-4895.
[3] M. Baser, A. Harmanci, T. K. Kwak, Generalized semicommutative rings and their extensions, Bull. Korean. Math. Soc. 45 (2008), 285-297.
[4] M. Baser, C. Y. Hong, T. K. Kwak, On extended reversible rings, Algebra. Colloq. 16 (2009), 37-48.
[5] M. Baser, F. Kaynabca, T. K. Kwak, Ring endomorphism with the reversible condition, Comm. Korean Math. Soc. 25 (2010), 349-364.
[6] P. M. Cohn, Reversible rings, Bull. London. Math. Soc. 31 (1999), 641-648.
[7] J. L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (1932), 85-88.
[8] E. Hashemi, A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta. Math. Hungar. 107 (2005), 207-224.
[9] C. Y. Hong, N. K. Kim, T. K. Kwak, Ore extension of Baer and p.p.-rings, J. Pure. Appl. Algebra. 151 (2000), 215-226.
[10] S. U. Hwang, Y. C. Jeon, Y. Lee, Structure and topological conditions of NI rings, J. Algebra. 302 (2006), 186-199.
[11] D. A. Jordan, Bijective extensions of injective ring endomorphisms, J. Lond. Math. Soc. 25 (1982), 435-448.
[12] N. K. Kim, T. K. Kwak, Y. Lee, Insertion-of-Factors-Property skewed by ring endomorphisms, Taiwan. J. Math. 18 (2014), 849-869.
[13] N. K. Kim, Y. Lee, Extensions of reversible rings, J. Pure. Appl. Algebra. 185 (2003), 207-223.
[14] J. Krempa, Some example of reduced rings, Algebra. Colloq. 3 (1996), 289-300.
[15] G. Marks, On 2-primal Ore extensions, Comm. Algebra. 29 (2001), 2113-2123.
[16] A. R. Nasr-Isfahani, A. Moussavi, On skew power serieswise Armendariz rings, Comm. Algebra. 39 (2011), 3114-3132.
