ON φ-PSEUDO ALMOST VALUATION RINGS
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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.
Abstract
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.
참고문헌 (18)
- D. F. Anderson, "On ϕ-Pr¨ufer rings and ϕ-B´ezout rings", Houston J. Math, 30(2), pp.331~343 (2004)
- D. F. Anderson, "Pairs of rings with the same prime ideals", Canad. J. Math, 32(2), pp.362~384 (1980)
- D. F. Anderson, "Idealization of a module", J. Commut. Algebra, 1(1), pp.3~56 (2009)
- D. F. Anderson, "Almost B´ezout domains", J. Algebra, 142(2), pp.285~309 (1991)
- M. F. Atiyah, "Introduction to Commutative Algebra" (1969), Addition-Wesley Publishing Company
- A. Badawi, "On divided rings and ϕ-pseudo-valuation rings, Commutative ring theory (F´es, 1995)", Lecture Notes in Pure and Appl. Math, 185, pp.57~67 (1997), Dekker
- A. Badawi, "On ϕ-pseudo-valuation rings, in Advances in Commutative Ring Theory, (Fez, Morocco 1997)", Lecture Notes Pure Appl. Math, 205, pp.101~110 (1999)
- A. Badawi, "On divided commutative rings", Comm. Algebra, 27(3), pp.1465~1474 (1999)
- A. Badawi, "On ϕ-pseudo-valuation rings II", Houston J. Math, 26(3), pp.473~480 (2000)
- A. Badawi, "On ϕ-chained rings and ϕ-pseudo-valuation rings", Houston J. Math, 27(4), pp.725~736 (2001)
- A. Badawi, "On Nonnil-Noetherian rings", Comm. Algebra, 31(4), pp.1669~1677 (2003)
- A. Badawi, "On pseudo almost valuation domains", Comm. Algebra, 35(4), pp.1167~1181 (2007)
- A. Badawi, "Pseudo-valuation rings", Lecture Notes Pure Appl. Math, 185, pp.57~67 (1997), Marcel Dekker
- G. W. Chang, "Generalizations of pseudo-valuation rings, Commutative rings", pp.15~24 (2002), Nova Sci. Publ
- D. E. Dobbs, "Divided rings and going-down", Pacific J. Math, 67(2), pp.353~363 (1976)
- J. R. Hedstrom, "Pseudo-valuation domains", Pacific J. Math, 75(1), pp.137~147 (1978)
- J. R. Hedstrom, "Pseudo-valuation domains II", Houston J. Math, 4(2), pp.199~207 (1978)
- I. Kaplansky, "Commutative Rings" (1974), Univ. Chicago Press