Summary: We recall that a ring \(R\) is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal.
Let \(R\) be a commutative ring, \(\sigma\) an automorphism of \(R\). Recall that a prime ideal \(P\) of \(R\) is \(\sigma\)-divided if it is comparable (under inclusion) to every \(\sigma\)-stable ideal \(I\) of \(R\). A ring \(R\) is called a \(\sigma\)-divided ring if every prime ideal of \(R\) is \(\sigma\)-divided. Also a ring \(R\) is almost \(\sigma\)-divided ring if every minimal prime ideal of \(R\) is \(\sigma\)-divided.
We also recall that a prime ideal \(P\) of \(R\) is \(\delta\)-divided if it is comparable (under inclusion) to every \(\delta\)-invariant ideal \(I\) of \(R\). A ring \(R\) is called a \(\delta\)-divided ring if every prime ideal of \(R\) is \(\delta\)-divided. A ring \(R\) is said to be almost \(\delta\)-divided ring if every minimal prime ideal of \(R\) is \(\delta\)-divided.
We define a Min.Spec-type endomorphism \(\sigma\) of a ring \(R\) (\(\sigma(U)\subseteq U\) for all minimal prime ideals \(U\) of \(R\)) and a Min.Spec-type ring (if there exists a Min.Spec-type endomorphism of \(R\)).
With this we prove the following. Let \(R\) be a commutative Noetherian \(\mathbb{Q}\)-algebra (\(\mathbb{Q}\) is the field of rational numbers), \(\delta\) a derivation of \(R\). Then: (1) \(R\) is a near pseudo valuation ring implies that \(R[x;\delta]\) is a near pseudo valuation ring. (2) \(R\) is an almost \(\delta\)-divided ring if and only if \(R[x;\delta]\) is an almost \(\delta\)-divided ring. – We also prove a similar result for \(R[x;\sigma]\), where \(R\) is a commutative Noetherian ring and \(\sigma\) a Min.Spec-type automorphism of \(R\).