IJPAM: Volume 52, No. 5 (2009)

PRIME RADICAL OF ORE EXTENSIONS OVER $\delta$-RINGS

V.K. Bhat$^1$, Ravi Raina$^2$
$^{1,2}$School of Mathematics
Shri Mata Vaishno Devi University (SMVD University)
Katra (J and K), 182320, INDIA
$^1$e-mail: vijaykumarbhat2000@yahoo.com
$^2$e-mail: rainaravirr@yahoo.co.in


Abstract.Let R be a ring. Let $\sigma$ be an automorphism of R. We recall the definition of a $\sigma(*)$-ring, and find a relation between the prime radical of a $\sigma(*)$-ring R and that of $R[x;\sigma]$. Let now $\delta$ be a $\sigma$-derivation of R. We say that a ring R is a $\delta$-ring if $a\delta(a)\in P(R)$ implies $a\in P(R)$, $a \in R$; where P(R) is the prime radical of R. We then find a relation between the prime radical of a $\delta$-ring R and that of $R[x;\sigma;\delta]$. We generalize the result for a Noetherian Q-algebra (Q is the field of rational numbers).

Received: July 26, 2007

AMS Subject Classification:

Key Words and Phrases: radical, automorphism, derivation, completely prime, $\delta$-ring, Q-algebra

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 5