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

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.

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### References:

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