On a generalization of central Armendariz rings
الموضوعات :
M. Sanaei
1
(Department of Mathematics, Islamic Azad University, Central Tehran Branch, 13185/768, Iran)
Sh. Sahebi
2
(Department of Mathematics, Islamic Azad University, Central Tehran Branch, 13185/768, Iran)
H. H. S. Javadi
3
(Department of Mathematics and Computer Science, Shahed University, Tehran, Iran)
الکلمات المفتاحية: central Armendariz rings, $alpha$-skew Armendariz rings, central $alpha$-skew Armendariz rings,
ملخص المقالة :
In this paper, some properties of $\alpha$-skew Armendariz and central Armendariz rings have been studied by variety of others. We generalize the notions to central $\alpha$-skew Armendariz rings and investigate their properties. Also, we show that if $\alpha(e)=e$ for each idempotent $e^{2}=e \in R$ and $R$ is $\alpha$-skew Armendariz, then $R$ is abelian. Moreover, if $R$ is central $\alpha$-skew Armendariz, then $R$ is right p.p-ring if and only if $R[x;\alpha]$ is right p.p-ring. Then it is proved that if $\alpha^{t}=I_{R}$ for some positive integer $t$, $ R $ is central $ \alpha $-skew Armendariz if and only if the polynomial ring $ R[x] $ is central $ \alpha $-skew Armendariz if and only if the Laurent polynomial ring $R[x,x^{-1}]$ is central $\alpha$-skew Armendariz.‎
[1] N. G. Agayev, G. Harmanci, A. Halicioglu, Central Armendariz rings, Bull. Malaysian. Math. Sci. Soc. 34 (1) (2011), 137-145.
[2] D. D. Anderson, V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra. 26 (7) (1998), 2265-2272.
[3] E. P. Armendariz, A note on extensions of Baer and p.p-rings, J. Austral. Math. Soc. 18 (1974), 470-473.
[4] W. Chen, W. Tong, A note on skew Armendariz rings, Comm. Algebra. 33 (4) (2005), 1137-1140.
[5] W. Chen, W. Tong, Ring endomorphisms with the reversible condition, Commun. Korean. Math. Soc. 25 (3) (2010), 349-364.
[6] Ch. Y. Hong, N. k. Kim, T. K. Kwak, On skew Armendariz rings, Comm. Algebra. 31 (1) (2003), 103-122.
[7] Ch. Y. Hong, T. K. Kwak, S. T. Rizivi, Extention of generalized Armendariz rings, Algebra. Colloq. 13 (2) (2006), 253-266.
[8] C. Huh, Y. Lee, A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra. 30 (2) (2002), 751-761.
[9] N. K. Kim, Y. Lee, Armendariz rings and reduced rings, J. Algebra. 223 (2) (2000), 477-488.
[10] M. B. Rege, S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Math. Sci. (Ser A). 73 (1997), 14-17.
