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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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참고문헌 (18)
참고문헌 신청
A. Badawi / 1997 / On divided rings and ?-pseudo-valuation rings, Commutative ring theory (F´es, 1995) / Lecture Notes in Pure and Appl. Math 185 : 57 ~ 67
A. Badawi / 1997 / Pseudo-valuation rings / Lecture Notes Pure Appl. Math 185 : 57 ~ 67
A. Badawi / 1999 / On ?-pseudo-valuation rings, in Advances in Commutative Ring Theory, (Fez, Morocco 1997) / Lecture Notes Pure Appl. Math 205 : 101 ~ 110
A. Badawi / 1999 / On divided commutative rings / Comm. Algebra 27 (3) : 1465 ~ 1474
A. Badawi / 2000 / On ?-pseudo-valuation rings II / Houston J. Math 26 (3) : 473 ~ 480
A. Badawi / 1997 / Pseudo-valuation rings / Lecture Notes Pure Appl. Math 185 : 57 ~ 67
A. Badawi / 1999 / On ?-pseudo-valuation rings, in Advances in Commutative Ring Theory, (Fez, Morocco 1997) / Lecture Notes Pure Appl. Math 205 : 101 ~ 110
A. Badawi / 1999 / On divided commutative rings / Comm. Algebra 27 (3) : 1465 ~ 1474
A. Badawi / 2000 / On ?-pseudo-valuation rings II / Houston J. Math 26 (3) : 473 ~ 480
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