MINIMAL PRIME IDEALS OF SKEW POLYNOMIAL RINGS AND NEAR PSEUDO-VALUATION RINGS

摘要

Let R be a ring. We recall that R is called a near pseudo-valuation ring if every minimal prime ideal of R is strongly prime. Let now σ be an automorphism of R and δ a σ-derivation of R. Then R is said to be an almost δ-divided ring if every minimal prime ideal of R is δ-divided. Let R be a Noetherian ring which is also an algebra over Q (Q is the field of rational numbers). Let σ be an automorphism of R such that R is a σ(*)-ring and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a 2 R. Further, if for any strongly prime ideal U of R with σ(U) = U and δ(U)?δ, U[x; σ, δ] is a strongly prime ideal of R[x; σ, δ], then we prove the following: (1) R is a near pseudo valuation ring if and only if the Ore extension 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.