Abstract
For a monoid M, we introduce M-nil-Armendariz rings, which are generalizations of nil-Armendariz rings and M-Armendariz rings, and we investigate their properties. We show that every NI ring is M-nil-Armendariz for any unique product monoid M, and if R is a 2-primal and M-Armendariz ring, then R is M × N-nil-Armendariz, where N is a unique product monoid. Moreover, we study the relationship between the weak annihilator ideals of a ring R and those of the monoid ring R[M] in case R is M-nil-Armendariz.
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References
Anderson D.D., Camillo V.: Armendariz rings and Gaussian rings. Commun. Algebra 26(7), 2265–2275 (1998)
Antoine R.: Nilpotent elements in Armendariz rings. J. Algebra 319(8), 3128–3140 (2008)
Armendariz E.P.: A note on extensions of Baer and p.p.-rings. J. Aust. Math. Soc. 18, 470–473 (1974)
Birkenmeier, G.F.; Heatherly, H.E.; Lee, E.K.: Completely prime ideals and associative radicals. In: Tain, S.K.; Rizvi, S.T. (eds.) Proceedings of Biennial Ohio Statt-Denison Conference 1992, pp. 102–129. Springer, New Jersey (1993)
Birkenmeier G.F., Park J.K.: Triangular matrix representations of ring extensions. J. Algebra 265(2), 457–477 (2003)
Hong C.Y., Kim N.K., Kwak T.K.: On skew Armendariz rings. Commun. Algebra 31(1), 103–122 (2003)
Huh C., Lee Y., Smoktunowicz A.: Armendariz rings and semicommutative rings. Commun. Algebra 30(2), 751–761 (2002)
Huh C., Lee C.I., Park V.S., Ryu J.: On π-Armendariz rings. Bull. Korean Math. Soc. 44(4), 641–649 (2007)
Kim N.K., Lee Y.: Armendariz rings and reduced rings. J. Algebra 223, 477–488 (2000)
Lee T.K., Wong T.L.: On Armendariz rings. Houston J. Math. 29(3), 583–593 (2003)
Liu Z.K.: Armendariz rings relative to a monoid. Commun. Algebra 33, 649–661 (2005)
Matczuk J.: A characterization of σ-rigid rings. Commun. Algebra 32(11), 4333–4336 (2004)
Moussavi A., Hashemi E.: On (α,δ)-skew Armendariz rings. J. Korean Math. Soc. 42(2), 353–363 (2005)
Okninski J.: Semigroup Algebras. Marcel Dekker, New York (1991)
Ouyang L.: Extensions of nilpotent p.p.-rings. Bull. Iranian Math. Soc. 36(2), 169–184 (2010)
Passman D.S.: The Algebra Structure of Group Rings. Wiley, New York (1977)
Rege M.B., Chhawchharian S.: Armendariz rings. Proc. Jpn. Acad. Ser. A Math. Sci. 73, 14–17 (1997)
Ribenboim P.: Noetherian rings of generalized power series. J. Pure Appl. Algebra 79, 293–312 (1992)
Acknowledgments
We would like to thank the referees for their valuable comments. This research is supported by the National Natural Science Foundation of China (10771058, 11071062), Natural Science Foundation of Hunan Province (10jj3065) and Scientific Research Foundation of Hunan Provincial Education Department (10A033).
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