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Research articles

On 2-primal Ore extensions over Noetherian $\sigma(*)$-rings

Vijay Kumar Bhat

School of Mathematics, SMVD University, Katra, India

Abstract: In this article, we discuss the prime radical of skew polynomial rings over Noetherian rings. We recall $\sigma(*)$ property on a ring $R$ (i.e. $a\sigma(a)\in P(R)$ implies $a\in P(R)$ for $a\in R$, where $P(R)$ is the prime radical of $R$, and $\sigma$ an automorphism of $R$). Let now $\delta$ be a $\sigma$-derivation of $R$ such that $\delta(\sigma(a))=\sigma(\delta(a))$ for all $a\in R$. Then we show that for a Noetherian $\sigma(*)$-ring, which is also an algebra over $\mathbb Q$, the Ore extension $R[x;\sigma,\delta]$ is 2-primal Noetherian (i.e. the nil radical and the prime radical of $R[x;\sigma,\delta]$ coincide).

Keywords and phrases: minimal prime, 2-primal, prime radical, automorphism, derivation.

MSC: 16S36, 16N40, 16P40, 16S32, 16W20, 16W25

Received: 02.06.2010

Language: English

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