Armendariz Property Of Upper Triangular Matrix Rings

Let R be an associative ring with identity. A ring R is called Armendariz if, whenever (?) = 0 in R[x], a_ib_j =0 for all i and j. A ring R is called reduced if it has no non-zero nilpotent elements. Every reduced ring is Armendariz by Armendariz[2]. Some properties of Armendariz rings have been studied in Rege Chhawchharia[1], Anderson and Camllo[3], Kim and Lee[4], Huh et al.[6] and Lee and Wong[7].Hong et al.[11] have studied a generalization of Armendariz rings, which they calledα-skew Armendariz rings, whereαis an endomorphism of R. Jerzy Matczuk[22] has studiedα-skew Armendariz rings. Zhongkui Liu[13] has studied another generalization of Armendariz rings, which they called M-Armendariz rings, where M is a monoid of R.An endomorphismαof a ring R is called to be rigid if aα(a) = 0 impliesα= 0 for a∈R. A ring R is called to beα-rigid if there exists a rigid endomorphismαof R. R is called a skew Armendariz ring with the endomorphismα(simply,aα-skew Armendariz ring ) if for p = (?) and q= (?) in R[x;α],pq = 0 implies a_iα~i(b_j) = 0 for all 0≤i≤m and 0≤j≤n.Let M be a monoid. A ring R is called M-Armendariz if,whenever (?) (?) = 0 in R[M],a_ib_j = 0 for all i and j.In this paper, we continue the study of Armendariz property of matrix rings. First, we study mainly Armendariz property of upper triangular matrix rings over reduced ring, identify several classes of Armendariz subrings of upper triangular matrix rings over reduced ring, and three classes of Armendariz subrings of upper triangular matrix rings over reduced ring among them are maximal Armendariz subrings of upper triangular matrix rings over reduced ring, whice are a class of Armendariz subrings of (2k + 1)×(2k + 1) upper triangular matrix rings over reduced ring and two classes of Armendariz subrings of (2k)×(2k) upper triangular matrix rings over reduced ring, where k is a positive integer, which are a generation of Tsiu-Kwen Lee and Yiqiang Zhou[8, Theorem1.4 and Proposition1.7]; Second, we study mainlyα-skew Armendariz property of upper triangular matrix rings overα-rigid ring, identify several classes ofα-skew Armendariz subrings of upper triangular matrix rings overα-rigid ring, and three classes ofα-skew Armendariz subrings of upper triangular matrix rings overα-rigid ring among them are maximalα-skew Armendariz subrings of upper triangular matrix rings overα-rigid ring, whice are a class ofα-skew Armendariz subrings of (2k + 1)×(2k + 1) upper triangular matrix rings overα-rigid ring and two class of maximalα-skew Armendariz subrings of (2k)×(2k) upper triangular matrix rings overα-rigid ring, where k is a positive integer; Last, we study mainly M-Armendariz property of upper triangular matrix rings over M-Armendariz and reduced ring, identify several classes of M-Armendariz subrings of upper triangular matrix rings over M-Armendariz and reduced ring, and three classes of M-Armendariz subrings of upper triangular matrix rings over M-Armendariz and reduced ring among them are maximal M-Armendariz subrings of upper triangular matrix rings over M-Armendariz and reduced ring, whice are a class of M-Armendariz subrings of (2k +1)×(2k+1) upper triangular matrix rings over M-Armendariz and reduced ring and two class of maximal M- Armendariz subrings of (2k)×(2k) upper triangular matrix rings over M-Armendariz and reduced ring, where k is a positive integer.