Let M be a monoid. A ring R is called M-π-Armendariz if whenever a = a1g1＋ a292 ＋ …＋angn, β= b1h1 ＋ b2h2 ＋ …＋ bmhm ∈ R[M] satisfy αβ ∈ nil（R[M]）, then aibj ∈ nil（R） for all i, j. A ring R is called ...Let M be a monoid. A ring R is called M-π-Armendariz if whenever a = a1g1＋ a292 ＋ …＋angn, β= b1h1 ＋ b2h2 ＋ …＋ bmhm ∈ 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-Tr-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 RIM] in case R is M-π-Armendariz.展开更多
基金This work was supported by the National Natural Science Foundation of China （Grant No. 11071097） and the Natural Science Foundation of Jiangsu Province （BK20141476）.
文摘Let M be a monoid. A ring R is called M-π-Armendariz if whenever a = a1g1＋ a292 ＋ …＋angn, β= b1h1 ＋ b2h2 ＋ …＋ bmhm ∈ 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-Tr-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 RIM] in case R is M-π-Armendariz.
