RESEARCH ARETICLE

π-Armendariz rings relative to a monoid

  • Yao WANG 1 ,
  • Meimei JIANG 1 ,
  • Yanli REN , 2
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  • 1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
  • 2. School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China

Received date: 24 Sep 2015

Accepted date: 19 Apr 2016

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a1g1+ a2g2+ · · · + angn, β = b1h1+ b2h2+ · · · + bmhmR[M] satisfy αβ ∈ nil(R[M]), then aibj ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.

Cite this article

Yao WANG , Meimei JIANG , Yanli REN . π-Armendariz rings relative to a monoid[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 1017 -1036 . DOI: 10.1007/s11464-016-0561-8

1
Alhevaz A, Moussavi A, Habibi M. On rings having McCoy-like conditions. Comm Algebra, 2012, 40: 1195–1221

DOI

2
Anderson D D, Camillo V. Armendariz rings and Gaussian rings. Comm Algebra, 1998, 26: 2265–2272

DOI

3
Antoine R. Nilpotent elements in Armendariz rings. Comm Algebra, 2008, 319(8): 3128–3140

DOI

4
Chen W. On nil-semicommutative rings. Thai J Math, 2011, 9: 39–47

5
Chen W, Cui S. On weakly semicommutative rings. Commun Math Res, 2011, 27(2): 179–192

6
Hashemi E. Nil-Armendariz rings relative to a monoid. Mediterr J Math, 2013, 10: 111–121

DOI

7
Hong C Y, Kim N K, Kwak T K, Lee Y. Extensions of zip rings. J Pure Appl Algebra, 2005, 195: 231–242

DOI

8
Huh C, Lee C I, Park K S, Ryu S J. On π-Armendariz rings. Bull Korean Math Soc, 2007, 44(4): 641–649

DOI

9
Huh C, Lee Y, Smoktunowicz A. Armendariz rings and semicommutative rings. Comm Algebra, 2002, 30(2): 751–761

DOI

10
Hwang S U, Jeon Y C, Lee Y. Structure and topological conditions of NI rings. J Algebra, 2006, 302: 186–199

DOI

11
Kim N K, Lee Y. Armendariz rings and reduced rings. Comm Algebra, 2000, 223: 477–488

DOI

12
Liu Z. Armendariz rings relative to a monoid. Comm Algebra, 2005, 33: 649–661

DOI

13
Liu Z, Zhao R. On weak Armendariz rings. Comm Algebra, 2006, 34(7): 2607–2616

DOI

14
Marks G. On 2-primal Ore extensions. Comm Algebra, 2001, 29(5): 2113–2123

DOI

15
Ouyang L. Ore extensions of weak zip rings. Glasg Math J, 2009, 51: 525–537

DOI

16
Ouyang L. Extensions of nilpotent p.p.-rings. Bull Iranian Math Soc, 2010, 36(2): 169–184

17
Ouyang L. On weak annihilator ideals of skew monoid rings. Comm Algebra, 2011, 39: 4259–4272

DOI

18
Ouyang L, Liu J. On a generalization of the π-Armendariz condition. Int Math Forum, 2011, 6(67): 3349–3356

19
Ouyang L, Liu J. Nil-Armendariz rings relative to a monoid. Arab J Math, 2013, 2: 81–90

DOI

20
Ouyang L, Liu J. Weak annihilator property of Malcev-Neumann rings. Asian Acad Management J Accounting Finance, 2013, 9(2): 1–14

21
Rege M, Chhawchharia S. Armendariz rings. Proc Japan Acad Ser A Math Sci, 1997, 73(1): 14–17

DOI

22
Ribenboim P. Semisimple rings and von Neumann regular rings of generalized power series. Comm Algebra, 1997, 198: 327–338

DOI

23
Zhang C, Chen J. Weak M-Armendariz rings. J Southeast Univ, 2009, 25(1): 142–146

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