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ON φ-PSEUDO ALMOST VALUATION RINGS
Afsaneh Esmaeelnezhad,Parviz Sahandi 대한수학회 2015 대한수학회보 Vol.52 No.3
The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be a φ-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map φ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a φ-ring R is said to be a φ-pseudo-strongly prime ideal if, whenever x, y ∈ RNil(R) and (xy)φ(P) ⊆ φ(P), then there exists an integer m ≥ 1 such that either xm ∈ φ(R) or ymφ(P) ⊆ φ(P). If each prime ideal of R is a φ-pseudo strongly prime ideal, then we say that R is a φ-pseudo-almost valuation ring (φ-PAVR). Among the properties of φ-PAVRs, we show that a quasilocal φ-ring R with regular maximal ideal M is a φ-PAVR if and only if V = (M : M) is a φ-almost chained ring with maximal ideal √MV . We also investigate the overrings of a φ-PAVR.
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SOME RESULTS CONCERNING (θ, φ)-DERIVATIONS ON PRIME RINGS
PARK, KYOO-HONG,JUNG, YONG-SOO 西原大學校 基礎科學硏究所 2004 基礎科學硏究論叢 Vol.18 No.-
Let R be a prime ring with characteristic different from two and let θ, ψ, σ, τ be the automorphisms of R. Let d :R→R be a nonzero (θ,ψ)-derivation. We prove the following results: (ⅰ) if a ∈ R and [d(R),a]θoσ, ψoτ = 0, then (σ(a) +τ(o)∈Z, the center of R, (ⅱ) if d([R,a]σ, τ) = 0, then σ(a) + τ(a)∈Z, (ⅲ) if ([ad(x), x:]σ, τ = 0 for all x ∈R, then a = 0 or R is commutative.
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