On semi-pseudo-valuation rings and their extensions. (English) Zbl 1255.16024

Bhat, V. K.

Lobachevskii J. Math. 31, No. 1, 8-12 (2010).

Summary: Recall that a commutative ring \(R\) is said to be a pseudo-valuation ring (PVR) if every prime ideal of \(R\) is strongly prime. We say that a commutative Noetherian ring \(R\) is semi-pseudo-valuation ring if assassinator of every right ideal (which is uniform as a right \(R\)-module) is a strongly prime ideal. We also recall that a prime ideal \(P\) of a ring \(R\) is said to be divided if it is comparable (under inclusion) to every ideal of \(R\). A ring \(R\) is called a divided ring if every prime ideal of \(R\) is divided. Let \(R\) be a commutative ring, \(\sigma\) an automorphism of \(R\) and \(\delta\) a \(\sigma\)-derivation of \(R\). We say that a prime ideal \(P\) of \(R\) is \(\delta\)-divided if it is comparable (under inclusion) to every \(\sigma\)-stable and \(\delta\)-invariant ideal \(I\) of \(R\). A ring \(R\) is called a \(\delta\)-divided ring if every prime ideal of \(R\) is \(\delta\)-divided. We say that a Noetherian ring \(R\) is semi-\(\delta\)-divided ring if assassinator of every right ideal (which is uniform as a right \(R\)-module) is \(\delta\)-divided. Let \(R\) be a ring and \(\sigma\) an endomorphism of \(R\). Recall that \(R\) is said to be a \(\sigma(*)\)-ring if \(a\sigma(a)\in P(R)\) implies that \(a\in P(R)\), where \(P(R)\) is the prime radical of \(R\).
With this we prove the following: Let \(R\) be a semiprime commutative Noetherian \(\mathbb Q\)-algebra, \(\sigma\) an automorphism of \(R\) such that \(R\) is a \(\sigma(*)\)-ring and \(\delta\) a \(\sigma\)-derivation of \(R\). Then: (1) If any \(U \in\text{S.Spec}(R)\) with \(\sigma(U)=U\) and \(\delta(U)\subseteq U\) implies that \(O(U)\in\text{S.Spec}(O(R))\), then \(R\) is a semi-pseudo-valuation ring implies that \(R[x;\sigma,\delta]\) is a semi-pseudo-valuation ring. (2) If \(R\) is a semi-\(\delta\)-divided ring, then \(O(R)\) is a Noetherian semi-\(\delta\)-divided ring.

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
13F30	Valuation rings
16P40	Noetherian rings and modules (associative rings and algebras)
16D25	Ideals in associative algebras
16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)

Keywords:

Ore extensions; automorphisms; derivations; divided primes; pseudo-valuation rings; near-pseudo-valuation rings; semi-pseudo-valuation rings

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