IFP rings and near-IFP rings
2
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
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 need not be preserved by polynomial ring extensions. But it is shown that ∑ni=0 EaiE is a nonzero nilpotent ideal of E whenever R is an IFP ring and 0 ≠ f ∈ F is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and ai’s are the coefficients of f. We shall use the term near- IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.
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 need not be preserved by polynomial ring extensions. But it is shown that ∑ni=0 EaiE is a nonzero nilpotent ideal of E whenever R is an IFP ring and 0 ≠ f ∈ F is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and ai’s are the coefficients of f. We shall use the term near- IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.
참고문헌 (17)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bulletin of the Australian Mathematical Society, 2, pp.363~368 (1970)
- G. F. Birkenmeier, "Completely prime ideals and associated radicals, Ring theory (Granville, OH, 1992)", pp.102~129 (1993), World Sci. Publ.
- D. B. Erickson, "Orders for finite noncommutative rings", American Mathematical Monthly, 73, pp.376~377 (1966)
- K. R. Goodearl, "von Neumann Regular Rings, Monographs and Studies in Mathematics", pp.4~ (1979), Pitman
- K. Y. Ham, "On weak Armendariz rings", ~
- I. N. Herstein, "Topics in Ring Theory", The University of Chicago Press
- C. Y. Hong, "Rings whose nilpotent elements form a Levitzki radical ring", Communications in Algebra, 35(4), pp.1379~1390 (2007)
- C. Y. Hong, "On minimal strongly prime ideals", Communications in Algebra, 28(10), pp.4867~4878 (2000)
- C. Huh, "p.p. rings and generalized p.p. rings", Journal of Pure and Applied Algebra, 167(1), pp.37~52 (2002)
- C. Huh, "Prime radicals of formal power series rings", Bulletin of the Korean Mathematical Society, 4, pp.623~633 (2001)
- C. Huh, "A. Smoktunowicz, Armendariz rings and semicommutative rings", Communications in Algebra, 30(2), pp.751~761 (2002)
- S. U. Hwang, "Structure and topological conditions of NI rings", Journal of Algebra, 302(1), pp.186~199 (2006)
- N. K. Kim, "Extensions of reversible rings", Journal of Pure and Applied Algebra, 185(1), pp.207~223 (2003)
- A. A. Klein, "Rings of bounded index", Communications in Algebra, 12(1), pp.9~21 (1984)
- G. Marks, "On 2-primal Ore extensions", Communications in 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 (Perpignan, 1981),, pp.71~73 (1982)
- G. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Transactions of the American Mathematical Society, 184, pp.43~60 (1973)