On a ring property unifying reversible and right duo rings
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
The concepts of reversible, right duo, and Armendariz rings are known to play important roles in ring theory and they are independent of one another. In this note we focus on a concept that can unify them,calling it a right Armendarizlike ring in the process. We first find a simple way to construct a right Armendarizlike ring but not Armendariz (reversible, or right duo). We show the difference between right Armendarizlike rings and strongly right McCoy rings by examining the structure of right annihilators. For a regular ring R, it is proved that R is right Armendarizlike if and only if R is strongly right McCoy if and only if R is Abelian (entailing that right Armendarizlike, Armendariz, reversible, right duo, and IFP properties are equivalent for regular rings). It is shown that a ring R is right Armendarizlike, if and only if so is the polynomial ring over R, if and only if so is the classical right quotient ring (if any). In the process necessary (counter)examples are found or constructed.
Abstract
The concepts of reversible, right duo, and Armendariz rings are known to play important roles in ring theory and they are independent of one another. In this note we focus on a concept that can unify them,calling it a right Armendarizlike ring in the process. We first find a simple way to construct a right Armendarizlike ring but not Armendariz (reversible, or right duo). We show the difference between right Armendarizlike rings and strongly right McCoy rings by examining the structure of right annihilators. For a regular ring R, it is proved that R is right Armendarizlike if and only if R is strongly right McCoy if and only if R is Abelian (entailing that right Armendarizlike, Armendariz, reversible, right duo, and IFP properties are equivalent for regular rings). It is shown that a ring R is right Armendarizlike, if and only if so is the polynomial ring over R, if and only if so is the classical right quotient ring (if any). In the process necessary (counter)examples are found or constructed.
참고문헌 (31)
- D. D. Anderson, "Armendariz rings and Gaussian ring", Comm. Algebra, 26(7), pp.2265~2272 (1998)
- D. D. Anderson, "Semigroups and rings whose zero products commute", Comm. Algebra, 27(6), pp.2847~2852 (1999)
- R. Antoine, "Nilpotent elements and Armendariz rings", J. Algebra, 319(8), pp.3128~3140 (2008)
- E. P. Armendariz, "A note on extensions of Baer and P.P.-rings", J. Austral. Math. Soc, 18, pp.470~473 (1974)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc, 2, pp.363~368 (1970)
- V. Camillo, "McCoy rings and zero-divisors", J. Pure Appl. Algebra, 212(3), pp.599~615 (2008)
- P. M. Cohn, "Reversible rings", Bull. London Math. Soc, 31(6), pp.641~648 (1999)
- R. C. Courter, "Finite-dimensional right duo algebras are duo", Proc. Amer. Math. Soc, 84(2), pp.157~161 (1982)
- E. H. Feller, "Properties of primary noncommutative rings", Trans. Amer. Math. Soc, 89, pp.79~91 (1958)
- K. R. Goodearl, "Von Neumann Regular Rings" (1979), Pitman
- K. R. Goodearl, "An Introduction to Noncommutative Noetherian Rings" (1989), Cambridge University Press
- J. M. Habeb, "A note on zero commutative and duo rings", Math. J. Okayama Univ, 32, pp.73~76 (1990)
- Y. Hirano, "On annihilator ideals of a polynomial ring over a noncommutative ring", J. Pure Appl. Algebra, 168(1), pp.45~52 (2002)
- C. Y. Hong, "The McCoy condition on noncommu- tative rings", Comm. Algebra, 39(5), pp.1809~1825 (2011)
- C. Huh, "p.p. rings and generalized p.p. rings", J. Pure Appl. Algebra, 167(1), pp.37~52 (2002)
- C. Huh, "Armendariz rings and semicommutative rings", Comm. Algebra, 30(2), pp.751~761 (2002)
- N. K. Kim, "Armendariz rings and reduced rings", J. Algebra, 223(2), pp.477~488 (2000)
- N. K. Kim, "Extensions of reversible rings", J. Pure Appl. Algebra, 185(1-3), pp.207~223 (2003)
- T.-K. Lee, "On Armendariz rings", Houston J. Math, 29(3), pp.583~593 (2003)
- T.-K. Lee, "Armendariz and reduced rings", Comm. Algebra, 32(6), pp.2287~2299 (2004)
- G. Marks, "Reversible and symmetric rings", J. Pure Appl. Algebra, 174(3), pp.311~318 (2002)
- G. Marks, "A taxonomy of 2-primal rings", J. Algebra, 266(2), pp.494~520 (2003)
- G. Marks, "Duo rings and Ore extensions", J. Algebra, 280(2), pp.463~471 (2004)
- J. Matczuk, "Ore extensions over duo ring", J. Algebra, 297(1), pp.139~154 (2006)
- N. H. McCoy, "Remarks on divisors of zero", Amer. Math. Monthly, 49, pp.286~295 (1942)
- N. H. McCoy, "Annihilators in polynomial rings", Amer. Math. Monthly, 64, pp.28~29 (1957)
- L. Motais de Narbonne, "Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents", Proceedings of the 106th National Congress of Learned Societies, pp.71~73 (1981)
- P. P. Nielsen, "Semi-commutativity and the McCoy condition", J. Algebra, 298(1), pp.134~141 (2006)
- M. B. Rege, "Armendariz rings", Proc. Japan Acad. Ser. A Math. Sci, 73(1), pp.14~17 (1997)
- G. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Trans. Amer. Math. Soc, 184, pp.43~60 (1973)
- W. Xue, "Artinian duo rings and self-dualty", Proc. Amer. Math. Soc, 105(2), pp.309~313 (1989)