Korean J. Math. Vol. 22 No. 2 (2014) pp.279-288
DOI: https://doi.org/10.11568/kjm.2014.22.2.279

Reversibility over prime radicals

Article Sidebar

PDF
Published: Jun 15, 2014
Keywords:
quasi-reversible-over-prime-radical ring (QRPR ring), prime radical, 2-primal ring, semicommutative ring, NI ring, weakly semicommutative ring
Mathematics Subject Classification:
16D25, 16N40

Main Article Content

Da Woon Jung
Department of Mathematics, Pusan National University
Yang Lee
Department of Mathematics Education, Pusan National University
Hyo Jin Sung
Department of Mathematics, Pusan National University

Abstract

The studies of reversible and 2-primal rings have done important roles in noncommutative ring theory. We in this note introduce the concept of {\it quasi-reversible-over-prime-radical} (simply, {\it QRPR}) as a generalization of the 2-primal ring property. A ring is called {\it QRPR} if ab=0 for a,b∈R implies that ab is contained in the prime radical. In this note we study the structure of QRPR rings and examine the QRPR property of several kinds of ring extensions which have roles in noncommutative ring theory.



Article Details

References

[1] D.D. Anderson, V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27(1999), 2847-2852. Google Scholar

[2] D.D. Anderson and Camillo, Victor, Armendariz rings and gaussian rings, Comm. Algebra 26 (1998), 2265–2272. Google Scholar

[3] R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), 3128-3140. Google Scholar

[4] E.P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Aust. Math. Soc. 18 (1974), 470-473. Google Scholar

[5] H.E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. Google Scholar

[6] G.F. Birkenmeier, H.E. Heatherly, E.K. Lee, Completely prime ideals and asso- ciated radicals, Proc. Biennial Ohio State-Denison Conference 1992, edited by S. K. Jain and S. T. Rizvi, World Scientific, Singapore-New Jersey-London-Hong Kong (1993), 102-129. Google Scholar

[7] W. Chen, On nil-semicommutative rings, Thai J. M. 9 (2011), 39-47. Google Scholar

[8] P.M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), 641-648. Google Scholar

[9] N.J. Divinsky, Rings and Radicals, University of Toronto Press, Toronto, 1965. Google Scholar

[10] Y. Hirano, D.V. Huynh, J.K. Park, On rings whose prime radical contains all nilpotent elements of index two, Arch. Math. 66 Google Scholar

[11] (1996), 360-365. Google Scholar

[12] C. Huh, H. K. Kim and Y. Lee, P. P.-rings and generalized P. P.-rings, J. Pure Appl. Algebra, 167 (2002), 37-52. Google Scholar

[13] S.U. Hwang, Y.C. Jeon, Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), 186-199. Google Scholar

[14] N.K. Kim, Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477–488. Google Scholar

[15] T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. Google Scholar

[16] J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. Google Scholar

[17] L.Liang, L.M. Wang, Z.K. Liu, On a generalization of semicommutative rings, Taiwanese J, Math. 11 (2007) 1359-1368 Google Scholar

[18] G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), 2113-2123. Google Scholar

[19] 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 (Perpignan, 1981), Bib. Nat., Paris (1982), 71-73. Google Scholar

[20] M.B. Rege, S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14-17. Google Scholar

[21] L.H. Rowen, Ring Theory, Academic Press, Inc., San Diego, 1991. Google Scholar

[22] G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60. Google Scholar