Ore Extensions over near Pseudo-valuation Rings and Noetherian 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 and δ a σderivation of R. We recall that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-invariant and δ-invariant ideal I (i.e. σ(I) ⊆ I and δ(I) ⊆ 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. Recall that an endomorphism σ of a ring R is called Min.Spec-type if σ(U) ⊆ U for all minimal prime ideals U of R and R is 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), σ a Min.Spec-type automorphism of R and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Further let any strongly prime ideal U of R with σ(U) ⊆ U and δ(U) ⊆ U implies that U [x;σ, δ] is a strongly prime ideal of R[x;σ, δ]. Then (1) R is a near pseudo valuation ring implies that R[x;σ, δ] is a near pseudo valuation ring (2) R is an almost δ-divided ring if and only if R[x;σ, δ] is an almost δ-divided ring.
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Ore Extensions over Pseudo-valuation Rings
Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let δ be a derivation of R and σ be an automorphism of R. Then we prove the following: 1. If R is a Pseudo-valuation ring, then R[x, δ] is also a Pseudo-valuation ring. 2. If R is a divided ring, then R[x, δ] is also a divided ring. 3. If R is a Pseudo-valuation ring, thenR[x, x−1, σ] is also a Pseudo-valuation ri...
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