Title & Authors
Hashemi, Ebrahim;

Abstract
A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c $\small{{\in}}$ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/$\small{\sqrt{I}}$) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then $\small{T_n}$(R) and R[x]=($\small{x^n}$) are radically-symmetric, where ($\small{x^n}$) is the ideal of R[x] generated by $\small{x^n}$. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) ($\small{{\alpha}}$, $\small{{\delta}}$)-compatible ring, then R[x; $\small{{\alpha}}$, $\small{{\delta}}$] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].
Keywords
insertion of factors property;($\small{{\alpha}}$, $\small{{\delta}}$)-compatible ideals;$\small{{\alpha}}$-rigid ideals;Ore extensions;symmetric rings;semicommutative rings;
Language
English
Cited by
References
1.
D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.

2.
E. Hashemi, On ideals which have the weakly insertion of factors property, J. Sci. Islam. Repub. Iran 19 (2008), no. 2, 145-152.

3.
E. Hashemi, Compatible ideals and radicals of Ore extensions, New York J. Math. 12 (2006), 349-356.

4.
E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207-224.

5.
C. Y. Hong, N. Y. Kim, T. K. Kwak, and Y. Lee, Extensions of zip rings, J. Pure Appl. Algebra 195 (2005), no. 3, 231-242.

6.
C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Rigid ideals and radicals of Ore extensions, Algebra Colloq. 12 (2005), no. 3, 399-412.

7.
C. Huh, H. K. Kim, and Y. Lee, P.P.-rings and generalized P.P.-rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52.

8.
C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761.

9.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.

10.
N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223.

11.
J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.

12.
J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), no. 3, 359-368.

13.
L. Liang, L.Wang, and Z. Liu, On a generalization of semicommutative rings, Taiwanese J. Math. 11 (2007), no. 5, 1359-1368.

14.
G. Mason, Re exive ideals, Comm. Algebra 9 (1981), no. 17, 1709-1724.