On Rings Having McCoy-Like Conditions

摘要

In [4141. Nielsen , P. P. ( 2006 ). Semi-commutativity and the McCoy condition . J. Algebra 298 : 134 - 141 .[CrossRef], [Web of Science ®]View all references], Nielsen proves that all reversible rings are McCoy and gives an example of a semicommutative ring that is not right McCoy. At the same time, he also shows that semicommutative rings do have a property close to the McCoy condition. In this article we study weak McCoy rings as a common generalization of McCoy rings and weak Armendariz rings. Relations between the weak McCoy property and other standard ring theoretic properties is considered. We also study the weak skew McCoy condition, a generalization of the standard weak McCoy condition from polynomials to skew polynomial rings. We resolve the structure of weak skew McCoy rings and obtain various necessary or sufficient conditions for a ring to be weak skew McCoy, unifying and generalizing a number of known McCoy-like conditions in the special cases. Constructing various examples, we classify how the weak McCoy property behaves under various ring extensions. As a consequence we extend and unify several known results related to McCoy rings and Armendariz rings [66. Başer , M. , Kwak , T. K. , Lee , Y. ( 2009 ). The McCoy condition on skew polynomial rings . Comm. Algebra 37 ( 11 ): 4026 - 4037 .[Taylor & Francis Online], [Web of Science ®]View all references, 3535. Liu , Z. K., Zhao , R. Y. (2006). On weak Armendariz rings. Comm. Algebra 34(7):2607-2616.[Taylor & Francis Online], [Web of Science ®]View all references, 3838. Moussavi , A. , Hashemi , E. ( 2005 ). On (α, δ)-skew Armendariz rings . J. Korean Math. Soc. 42 ( 2 ): 353 - 363 .[CrossRef], [Web of Science ®]View all references, 4343. Ouyang , L. ( 2009 ). Ore extensions of weak zip rings . Glasgow Math. J. 51 : 525 - 537 .[CrossRef], [Web of Science ®]View all references, 4949. Zhang , C. P. , Chen , J. L. ( 2010 ). Weak α-Skew Armendariz rings . J. Korean Math. Soc. 47 ( 3 ): 455 - 466 .[CrossRef], [Web of Science ®]View all references].View full textDownload full textKey WordsMonoid ring, Semicommutative ring, Skew polynomial ring, Weak Armendariz ring, Weak McCoy ring, Weak zip ring2000 Mathematics Subject Classification16D15, 16D40, 16D70Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/00927872.2010.548842