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**On near pseudo-valuation rings and their extensions.**
*(English)*
Zbl 1162.13304

Int. Electron. J. Algebra 5, 70-77 (2009); corrigendum ibid. 134-135 (2009).

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 ring \(R\) is near pseudo-valuation ring if every minimal prime ideal 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 ring \(R\) is almost \(\delta\)-divided ring if every minimal prime ideal of \(R\) is \(\delta\)-divided. With this we prove the following:

Let \(R\) be a commutative noetherian \(\mathbb{Q}\)-algebra, \(\sigma\) and \(\delta\) as usual. Then:

(1) If \(R\) is a near pseudo valuation \(\delta(*)\)- ring, then \(R[x; \sigma, \delta]\) is a near pseudo valuation ring.

(2) If \(R\) is an almost \(\delta\)-divided \(\sigma(*)\)-ring, then \(R[x; \sigma, \delta]\) is an almost divided ring.

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 ring \(R\) is almost \(\delta\)-divided ring if every minimal prime ideal of \(R\) is \(\delta\)-divided. With this we prove the following:

Let \(R\) be a commutative noetherian \(\mathbb{Q}\)-algebra, \(\sigma\) and \(\delta\) as usual. Then:

(1) If \(R\) is a near pseudo valuation \(\delta(*)\)- ring, then \(R[x; \sigma, \delta]\) is a near pseudo valuation ring.

(2) If \(R\) is an almost \(\delta\)-divided \(\sigma(*)\)-ring, then \(R[x; \sigma, \delta]\) is an almost divided ring.