ANNIHILATORS IN ONE-SIDED IDEALS GENERATED BY COEFFICIENTS OF ZERO-DIVIDING POLYNOMIALS Kwak, Tai Keun; Lee, Dong Su; Lee, Yang;
Nielsen and Rege-Chhawchharia called a ring R right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, there exists a nonzero element r R with f(x)r = 0. Hong et al. called a ring R strongly right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, f(x)r = 0 for some nonzero r in the right ideal of R generated by the coefficients of g(x). Subsequently, Kim et al. observed similar conditions on linear polynomials by finding nonzero r's in various kinds of one-sided ideals generated by coefficients. But almost all results obtained by Kim et al. are concerned with the case of products of linear polynomials. In this paper we examine the nonzero annihilators in the products of general polynomials.
right left-ideal-McCoy ring;right McCoy ring;polynomial ring;matrix ring;condition ();Dorroh extension;
D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.
D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852.
E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Aust. Math. Soc. 18 (1974), 470-473.
V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599-615.
P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648.
K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 1989.
C. Y. Hong, Y. C. Jeon, N. K. Kim, and Y. Lee, The McCoy condition on noncommu- tative rings, Comm. Algebra 39 (2011), no. 5, 1809-1825.
C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761.
Y. C. Jeon, H. K. Kim, N. K. Kim, T. K. Kwak, Y. Lee, and D. E. Yeo, On a general-ization of the McCoy condition, J. Korean Math. Soc. 47 (2010), no. 6, 1269-1282.
L. G. Jones and L. Weiner, Advanced Problems and Solutions, Solutions 4419, Amer. Math. Monthly 59 (1952), no. 5, 336-337.
B. O. Kim, T. K. Kwak, and Y. Lee, On constant zero-divisors of linear polynomials, Comm. Algebra (to appear).
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.
N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223.
N. K. Kim, Y. Lee, and Y. Seo, Structure of idempotents in rings, (submitted).
T. K. Kwak and Y. Lee, Rings over which coefficients of nilpotent polynomials are nilpotent, Internat. J. Algebra Comput. 21 (2011), no. 5, 745-762.
T. K. Kwak, Y. Lee, and S. J. Yun, The Armendariz property on ideals, J. Algebra 354 (2012), 121-135.
J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.
J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368.
T. K. Lee and Y. Q. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (2004), no. 6, 2287-2299.
N. H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly 49 (1942), 286-295.
L. Motais de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, In: Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris, 1982.
P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006), no. 1, 134-141.
V. S. Ramamurthi, Weakly regular rings, Canad. Math. Bull. 16 (1973), 317-321.
M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17.
L. Xu and W. Xue, Structure of minimal non-commutative zero-insertive rings, Math. J. Okayama Univ. 40 (1998), 69-76.
W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings, Comm. Algebra 20 (1992), no. 9, 2777-2788.