WEAK α-SKEW ARMENDARIZ RINGS

Title & Authors
WEAK α-SKEW ARMENDARIZ RINGS
Zhang, Cuiping; Chen, Jianlong;

Abstract
For an endomorphism $\small{\alpha}$ of a ring R, we introduce the weak $\small{\alpha}$-skew Armendariz rings which are a generalization of the $\small{\alpha}$-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak $\small{\alpha}$-skew Armendariz if and only if for any n, the $\small{n\;{\times}\;n}$ upper triangular matrix ring $\small{T_n(R)}$ is weak $\small{\bar{\alpha}}$-skew Armendariz, where $\small{\bar{\alpha}\;:\;T_n(R)\;{\rightarrow}\;T_n(R)}$ is an extension of $\small{\alpha}$ If R is reversible and $\small{\alpha}$ satisfies the condition that ab = 0 implies $\small{a{\alpha}(b)=0}$ for any a, b $\small{\in}$ R, then the ring R[x]/($\small{x^n}$) is weak $\small{\bar{\alpha}}$-skew Armendariz, where ($\small{x^n}$) is an ideal generated by $\small{x^n}$, n is a positive integer and $\small{\bar{\alpha}\;:\;R[x]/(x^n)\;{\rightarrow}\;R[x]/(x^n)}$ is an extension of $\small{\alpha}$. If $\small{\alpha}$ also satisfies the condition that $\small{{\alpha}^t\;=\;1}$ for some positive integer t, the ring R[x] (resp, R[x; $\small{\alpha}$) is weak $\small{\bar{\alpha}}$-skew (resp, weak) Armendariz, where $\small{\bar{\alpha}\;:\;R[x]\;{\rightarrow}\;R[x]}$ is an extension of $\small{\alpha}$.
Keywords
reversible rings;$\small{\alpha}$-skew Armendariz rings;weak Armendariz rings;weak $\small{\alpha}$-skew Armendariz rings;
Language
English
Cited by
1.
On Rings Having McCoy-Like Conditions, Communications in Algebra, 2012, 40, 4, 1195
References
1.
D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.

2.
E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473.

3.
J. L. Chen and Y. Q. Zhou, Extensions of injectivity and coherent rings, Comm. Algebra 34 (2006), no. 1, 275-288.

4.
C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), no. 1, 103-122.

5.
C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761.

6.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.

7.
N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223.

8.
Z. K. Liu and R. Y. Zhao, On weak Armendariz rings, Comm. Algebra 34 (2006), no. 7, 2607-2616.

9.
M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17.