The Relation Between Units and Nilpotents
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We discuss the relation between units and nilpotents of a ring, concentrating on the transitivity of units on nilpotents under regular group actions. We first prove that for a ring R, if U(R) is right transitive on N(R), then Kothe’s onjecture holds for R, where U(R) and N(R) are the group of all units and the set of all nilpotents in R, respectively. A ring is called right UN-transitive if it satisfies this transitivity, as a generalization, a ring is called unilpotent-IFP if aU(R) ⊆ N(R) for all a ∈ N(R). We study the structures of right UN-transitive and unilpotent-IFP rings in relation to radicals, NI rings, unit-IFPrings, matrix rings and polynomial rings.
Abstract
We discuss the relation between units and nilpotents of a ring, concentrating on the transitivity of units on nilpotents under regular group actions. We first prove that for a ring R, if U(R) is right transitive on N(R), then Kothe’s onjecture holds for R, where U(R) and N(R) are the group of all units and the set of all nilpotents in R, respectively. A ring is called right UN-transitive if it satisfies this transitivity, as a generalization, a ring is called unilpotent-IFP if aU(R) ⊆ N(R) for all a ∈ N(R). We study the structures of right UN-transitive and unilpotent-IFP rings in relation to radicals, NI rings, unit-IFPrings, matrix rings and polynomial rings.
참고문헌 (16)
- R. Antoine, "Nilpotent elements and Armendariz rings", J. Algebra, 319, pp.3128~3140 (2008)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc, 2, pp.363~368 (1970)
- H. E. Bell, "Duo group rings", J. Pure Appl. Algebra, 209, pp.833~838 (2007)
- J. L. Dorroh, "Concerning adjunctins to algebras", Bull. Amer. Math. Soc, 38, pp.85~88 (1932)
- K. R. Goodearl, "Von Neumann Regular Rings" (1979), Pitman
- 한준철, "DUO RING PROPERTY RESTRICTED TO GROUPS OF UNITS", 대한수학회지, 52(3), pp.489~501 (2015), 대한수학회
- C. Huh, "Armendariz rings and semicommutative rings", Comm. Algebra, 30, pp.751~761 (2002)
- S. U. Hwang, "Structure and topological conditions of NI rings", J. Algebra, 302, pp.186~199 (2006)
- N. Jacobson, "Some remarks on one-sided inverses", Proc. Amer. Math. Soc, 1, pp.352~355 (1950)
- D. W. Jung, "Nil-Armendariz rings and upper nilradicals", Internat. J. Algebra Comput, 22, pp.1250059~ (2012)
- H. K. Kim, "Insertion of units at zero products", J. Algebra Appl, 17, pp.1850043~ (2018)
- 김남균, "SEMICOMMUTATIVE PROPERTY ON NILPOTENT PRODUCTS", 대한수학회지, 51(6), pp.1251~1267 (2014), 대한수학회
- 이양, "Structure of unit-IFP rings", 대한수학회지, 55(5), pp.1257~1268 (2018), 대한수학회
- G. Marks, "On 2-primal Ore extensions", Comm. Algebra, 29, pp.2113~2123 (2001)
- M. B. Rege, "Armendariz rings", Proc. Japan Acad. Ser. A Math. Sci, 73, pp.14~17 (1997)
- A. Smoktunowicz, "Polynomial rings over nil rings need not be nil", J. Algebra, 233, pp.427~436 (2000)