π-Armendariz rings relative to a monoid

doi:10.1007/s11464-016-0561-8

Front. Math. China

2016, Vol. 11

Issue (4) : 1017-1036 https://doi.org/10.1007/s11464-016-0561-8

RESEARCH ARETICLE

π-Armendariz rings relative to a monoid

Yao WANG¹,Meimei JIANG¹,Yanli REN^2,^*(

)

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

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Abstract

Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a₁g₁+ a₂g₂+ · · · + a_ng_n, β = b₁h₁+ b₂h₂+ · · · + b_mh_m ∈ R[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.

Keywords Monoid ring π-Armendariz ring M-π-Armendariz ring weakly 2-primal ring weak annihilator

Corresponding Author(s): Yanli REN

Issue Date: 30 August 2016

Cite this article:

Yao WANG,Meimei JIANG,Yanli REN. π-Armendariz rings relative to a monoid[J]. Front. Math. China, 2016, 11(4): 1017-1036.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0561-8
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I4/1017

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