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Vietnam Journal of Mathematics 36:4(2008) 455-461

Ore Extensions over 2-primal Rings

V. K. Bhat and Ravi Raina

Abstract. Let $R$ be a ring, $\sigma$ an automorphism of $R$ and let $\delta$ be a $\sigma$-derivation of $R$. Recall that a ring $R$ is said to be a $\delta$-ring if $a\delta(a)\in P(R)$ implies $a\in P(R)$, where $P(R)$ denotes the prime radical of $R$. \indent It is known that if $R$ is a $\delta$-Noetherian $Q$-algebra, $\sigma$ and $\delta$ are as usual such that $\sigma(\delta(a))=\delta(\sigma(a))$, for all $a\in R$ and $\sigma(P) = P$, for all minimal prime ideals $P$ of $R$, then $R[x,\sigma,\delta]$ is a 2-primal Noetherian ring. In this article it is proved that in the case $\delta$ is the zero map, $R$ is a 2-primal Noetherian ring implies that $R[x,\sigma]$ is a 2-primal Noetherian ring. In the case $\sigma$ is the identity map, a similar result is proved for the differential operator ring $R[x,\delta]$ ($R$ in this case is moreover a Noetherian $Q$-algebra).

1991 Mathematics Subject Classification: Primary 16XX, Secondary 16N40, 16P40, 16W20, 16W25.

Keywords: 2-primal, Minimal prime, prime radical, nil radical, automorphism, derivation.