IJPAM: Volume 70, No. 1 (2011)

ON REDUCED AND $\alpha$-SKEW QUASI-ARMENDARIZ MODULE

Prachi Juyal$^1$, M.R. Khan$^2$, V.N. Dixit$^3$
$^{1,2,3}$Department of Mathematics
Jamia Millia Islamia (Central University)
New Delhi, 110 025, INDIA


Abstract. Let $R$ be an associated ring with identity. A module $M_R$ be an $\alpha$-weakly rigid module which is generalization of $\alpha$-compatible module, where $\alpha$ is an monomorphism on $R$. In this paper we proved the following results. Let $M_R$ be a $\alpha$-weakly rigid module then the following conditions hold:

(a)
(i)
If $M[x, \alpha]_{R[x,\alpha]}$ is p.q. Baer then $M_R$ is p.q. Baer, converse is true if $M_R$ is $\alpha$-reduced.
(ii)
If $M[[x, \alpha]]_{R[[x, \alpha]]}$ is p.q. Baer then $M_R$ is p.q. Baer
(b)
(i)
If $M[x, x^{-1}, \alpha]_{R[x,x^{-1}, \alpha]}$ is p.q. Baer, Then $M_R$ is p.q. Baer converse is true if $M_R$ is $\alpha$-reduced
(ii)
If $M[[x, x^{-1},\alpha]]_{R[[x, x^{-1}, \alpha]]}$ is p.q. Baer, then $M_R$ is p.q. Baer
(c)
If $M_R$ is p.q. Baer, then $M_R$ is $\alpha$-skew quasi Armendariz module of power series type.
(d)
If $M_R$ is p.q. Baer, then $M_R$ is $\alpha$-skew quasi-Armendariz module of Laurent power series type.


Received: February 4, 2011

AMS Subject Classification: 16D80, 16S36, 16W60

Key Words and Phrases: $\alpha$-weakly rigid ring, $\alpha$-skew quasi-Armendariz module

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 1