Ore Extensions over near Pseudo-valuation Rings
نویسنده
چکیده
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, σ an automorphism of R. Recall that a prime ideal P of R is σ-divided if it is comparable (under inclusion) to every σ-stable ideal I of R. A ring R is called a σ-divided ring if every prime ideal of R is σ-divided. Also a ring R is almost σ-divided ring if every minimal prime ideal of R is σ-divided. We also recall that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every δ-invariant ideal I of R. A ring R is called a δ-divided ring if every prime ideal of R is δ-divided. A ring R is said to be almost δ-divided ring if every minimal prime ideal of R is δ-divided. We define a Min.Spec-type endomorphism σ of a ring R (σ(U) ⊆ 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 Q-algebra (Q is the field of rational numbers), δ a derivation of R. Then: (1) R is a near pseudo valuation ring implies that R[x; δ] is a near pseudo valuation ring. (2) R is an almost δ-divided ring if and only if R[x; δ] is an almost δdivided ring. We also prove a similar result for R[x; σ], where R is a commutative Noetherian ring and σ a Min.Spec-type automorphism of R.
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