RIGIDNESS AND EXTENDED ARMENDARIZ PROPERTY

Title & Authors
RIGIDNESS AND EXTENDED ARMENDARIZ PROPERTY
Baser, Muhittin; Kaynarca, Fatma; Kwak, Tai-Keun;

Abstract
For a ring endomorphism of a ring R, Krempa called $\small{\alpha}$ rigid endomorphism if $\small{a{\alpha}(a)}$ = 0 implies a = 0 for a $\small{\in}$ R, and Hong et al. called R an $\small{\alpha}$-rigid ring if there exists a rigid endomorphism $\small{\alpha}$. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., $\small{\alpha}$-Armendariz rings and $\small{\alpha}$-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between $\small{\alpha}$-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an $\small{\alpha}$-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.
Keywords
reduced rings;skew polynomial rings;rigid rings;(extended) Armendariz rings;trivial extension;semiprime rings;semicommutative rings;
Language
English
Cited by
1.
On α-nilpotent elements and α-Armendariz rings, Journal of Algebra and Its Applications, 2015, 14, 05, 1550064
References
1.
E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473.

2.
W. Chen and W. Tong, A note on skew Armendariz rings, Comm. Algebra 33 (2005), no. 4, 1137-1140.

3.
W. Chen and W. Tong, On skew Armendariz rings and rigid rings, Houston J. Math. 33 (2007), no. 2, 341-353.

4.
C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), no. 3, 215-226.

5.
C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), no. 1, 103-122.

6.
C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Extensions of generalized Armendariz rings, Algebra Colloq. 13 (2006), no. 2, 253-266.

7.
A. A. M. Kamal, Some remarks on Ore extension rings, Comm. Algebra 22 (1994), no. 10, 3637-3667.

8.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.

9.
J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.

10.
T. K. Lee and T. L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), no. 3, 583-593.

11.
T. K. Lee and Y. Q. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (2004), no. 6, 2287-2299.

12.
M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17.