We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of...We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn（R） is weakly nil-clean, and to show that the endomorphism ring EndD（V） over a vector space VD is weakly nil-clean if and only if it is nil-clean or dim（V） = 1 with D Z3.展开更多
A ring R is said to be weakly semicommutative if for any a,b∈R, ab=0 implies aRb（？）Nil（R）,where Nil（R） is the set of all nilpotent elements in R. In this note,we clarify the relationship between weakly semicom...A ring R is said to be weakly semicommutative if for any a,b∈R, ab=0 implies aRb（？）Nil（R）,where Nil（R） is the set of all nilpotent elements in R. In this note,we clarify the relationship between weakly semicommutative rings and NI-rings by proving that the notion of a weakly semicommutative ring is a proper generalization of NI-rings.We say that a ring R is weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical,and prove that if R is a weakly 2-primal ring which satisfiesα-condition for an endomorphismαof R（that is,ab=0（？）aα（b）=0 where a,b∈R） then the skew polynomial ring R[x;α] is a weakly 2-primal ring,and that if R is a ring and I is an ideal of R such that I and R/I are both weakly semicommutative then R is weakly semicommutative. Those extend the main results of Liang et al.2007（Taiwan Residents J.Math.,11（5）（2007）, 1359-1368） considerably.Moreover,several new results about weakly semicommutative rings and NI-rings are included.展开更多
A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let α be an endomorph...A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let α be an endomorphism and δ an α- derivation of a ring R. We prove that （1） If R is an （α, δ）-compatible and weakly 2-primal ring, then R[x; α, δ] is weakly semicommutative; （2） If R is （α, δ）-compatible, then R is weakly 2-primal if and only if R[x; α, δ] is weakly 2-primal.展开更多
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
文摘We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn（R） is weakly nil-clean, and to show that the endomorphism ring EndD（V） over a vector space VD is weakly nil-clean if and only if it is nil-clean or dim（V） = 1 with D Z3.
基金The NSF(Y2008A04,ZR2010AM003,BS2010SF107) of Shandong Province,China
文摘A ring R is said to be weakly semicommutative if for any a,b∈R, ab=0 implies aRb（？）Nil（R）,where Nil（R） is the set of all nilpotent elements in R. In this note,we clarify the relationship between weakly semicommutative rings and NI-rings by proving that the notion of a weakly semicommutative ring is a proper generalization of NI-rings.We say that a ring R is weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical,and prove that if R is a weakly 2-primal ring which satisfiesα-condition for an endomorphismαof R（that is,ab=0（？）aα（b）=0 where a,b∈R） then the skew polynomial ring R[x;α] is a weakly 2-primal ring,and that if R is a ring and I is an ideal of R such that I and R/I are both weakly semicommutative then R is weakly semicommutative. Those extend the main results of Liang et al.2007（Taiwan Residents J.Math.,11（5）（2007）, 1359-1368） considerably.Moreover,several new results about weakly semicommutative rings and NI-rings are included.
基金The NSF(11071097,11101217)of Chinathe NSF(BK20141476)of Jiangsu Province
文摘A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let α be an endomorphism and δ an α- derivation of a ring R. We prove that （1） If R is an （α, δ）-compatible and weakly 2-primal ring, then R[x; α, δ] is weakly semicommutative; （2） If R is （α, δ）-compatible, then R is weakly 2-primal if and only if R[x; α, δ] is weakly 2-primal.
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.
