On conditions provided by Nilradicals
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A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for a, b∈ R. Huh et al. showed that the IFP condition is not preserved by polynomial ring extensions. In this note we concentrate on a generalized condition of the IFPness that can be lifted up to polynomial rings, introducing the concept of quasi-IFP rings. The structure of quasi-IFP rings will be studied, characterizing quasi-IFP rings via minimal strongly prime ideals. The connections between quasi-IFP rings and related concepts are also observed in various situations, constructing necessary examples in the process. The structure of minimal noncommutative (quasi-)IFP rings is also observed.
Abstract
A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for a, b∈ R. Huh et al. showed that the IFP condition is not preserved by polynomial ring extensions. In this note we concentrate on a generalized condition of the IFPness that can be lifted up to polynomial rings, introducing the concept of quasi-IFP rings. The structure of quasi-IFP rings will be studied, characterizing quasi-IFP rings via minimal strongly prime ideals. The connections between quasi-IFP rings and related concepts are also observed in various situations, constructing necessary examples in the process. The structure of minimal noncommutative (quasi-)IFP rings is also observed.
참고문헌 (25)
- A. Badawi, "On abelian ¼-regular rings", Comm. Algebra, 25(41), pp.1009~1021 (1997)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc., 2, pp.363~368 (1970)
- G. F. Birkenmeier, "Completely prime ideals and associated radicals, Ring theory", pp.102~129 (1993), World Sci. Publ.
- A. W. Chatters, "Rings with Chain Conditions, Research Notes in Mathematics", Pitman, 44 (1980)
- N. J. Divinsky, "Rings and Radicals, Mathematical Expositions", University of Toronto Press, (14) (1965)
- D. B. Erickson, "Orders for finite noncommutative rings", Amer. Math. Monthly, 73, pp.376~377 (1966)
- K. R. Goodearl, "von Neumann Regular Rings, Monographs and Studies in Mathematics", Pitman, 4 (1979)
- 함경연, "IFP rings and near-IFP rings", 대한수학회지, 45(3), pp.727~740 (2008), 대한수학회
- C. Y. Hong, "On minimal strongly prime ideals", Comm. Algebra, 28(10), pp.4867~4878 (2000)
- C. Huh, "p.p. rings and generalized p.p. rings", J. Pure Appl. Algebra, 167(1), pp.37~52 (2002)
- C. Huh, "Examples of strongly ¼-regular rings", J. Pure Appl. Algebra, 189(1), pp.195~210 (2004)
- C. Huh, "Armendariz rings and semicommutative rings", Comm. Algebra, 30(2), pp.751~761 (2002)
- S. U. Hwang, "Structure and topological conditions of NI rings", J. Algebra, 302(1), pp.186~199 (2006)
- S. U. Hwang, "On rings whose right annihilators are bounded", Glasgow Math. J., ~
- Young Cheol Jeon, "On weak Armendariz rings", 대한수학회보, 46(1), pp.135~146 (2009), 대한수학회
- N. K. Kim, "Extensions of reversible rings", J. Pure Appl. Algebra, 185(1), pp.207~223 (2003)
- N. K. Kim, "An ascending chain condition on Wedderburn radicals", Comm. Algebra, 34(1), pp.37~50 (2006)
- J. Lambek, "On the representation of modules by sheaves of factor modules", Canad. Math. Bull., 14, pp.359~368 (1971)
- C. Lanski, "Nil subrings of Goldie rings are nilpotent", Canad. J. Math., 21, pp.904~907 (1969)
- G. Marks, "On 2-primal Ore extensions", Comm. Algebra, 29(5), pp.2113~2123 (2001)
- L. Motais de Narbonne, "Anneaux semi-commutatifs et unis´eriels; anneaux dont les id´eaux principaux sont idempotents", Proceedings of the 106th National Congress of Learned Societies, pp.71~73 (1982), Bib. Nat.
- L. H. Rowen, "Ring Theory" (1991), Academic Press, Inc.
- G. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Trans. Amer. Math. Soc., 184, pp.43~60 (1974)
- W. Xue, "On strongly right bounded finite rings", Bull. Austral. Math. Soc., 44(3), pp.353~355 (1991)
- W. Xue, "Structure of minimal noncommutative duo rings and minimal strongly bounded nonduo rings", Comm. Algebra, 20(9), pp.2777~2788 (1992)