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Relative Ding Projective Modules over Formal Triangular Matrix Rings
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作者 Hongyan Fan Xi Tang 《Journal of Applied Mathematics and Physics》 2023年第6期1598-1614,共17页
Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using t... Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module. 展开更多
关键词 Formal triangular matrix ring Relative Ding Projective Module Relative Ding Projective Dimension
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Special Properties of Formal Triangular Matrix Rings 被引量:4
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作者 FAN Wei-li WANG Hui 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第4期625-632,共8页
We consider the sufficient and necessary conditions for the formal triangular matrix ring being right minsymmetric, right DS, semicommutative, respectively.
关键词 formal triangular matrix ring right minsymmetric ring DS ring semicommutative ring
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Uniquely strongly clean triangular matrix rings
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作者 崔建 陈建龙 《Journal of Southeast University(English Edition)》 EI CAS 2011年第4期463-465,共3页
An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of... An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of R is uniquely strongly clean. The uniquely strong cleanness of the triangular matrix ring is studied. Let R be a local ring. It is shown that any n × n upper triangular matrix ring over R is uniquely strongly clean if and only if R is uniquely bleached and R/J(R) ≈Z2. 展开更多
关键词 uniquely strongly clean ring uniquely bleached local ring triangular matrix ring
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Separativity and Stable Range for Formal Triangular Matrix Rings 被引量:3
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作者 黎奇升 佟文廷 《Northeastern Mathematical Journal》 CSCD 2003年第1期51-56,共6页
In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely gene... In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely generated projec-tive T-modules to have n in the stable range is given under the assumption that A and B are exchange rings. 展开更多
关键词 triangular matrix ring stable range condition SEPARABILITY
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On Skew Triangular Matrix Rings 被引量:3
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作者 M. Habibi A. Moussavi A. Alhevaz 《Algebra Colloquium》 SCIE CSCD 2015年第2期271-280,共10页
Let R be a ring with an endomorphism a. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S(R, n, σ) and T(R, n,σ). They allow the construction of... Let R be a ring with an endomorphism a. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S(R, n, σ) and T(R, n,σ). They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which inherit various interesting properties of rings. 展开更多
关键词 quasi-Armendariz ring clean ring triangular matrix ring
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Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings
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作者 Dejun Wu Hui Zhou 《Algebra Colloquium》 SCIE CSCD 2022年第3期475-490,共16页
Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-... Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied. 展开更多
关键词 formal triangular matrix ring Gorenstein AC-projective module level module absolutely clean module
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PS and CESS Property of Formal Triangular Matrix Rings
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作者 张文汇 刘仲奎 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2008年第4期981-986,共6页
Let R be a ring. Recall that a right R-module M (RR, resp.) is said to be a PS-module (PS-ring, resp.) if it has projective socle. M is called a CESS-module if every complement summand in M with essential socle is a d... Let R be a ring. Recall that a right R-module M (RR, resp.) is said to be a PS-module (PS-ring, resp.) if it has projective socle. M is called a CESS-module if every complement summand in M with essential socle is a direct summand of M. We show that the formal triangular matrix ring T = A 0M B is a PS-ring if and only if A is a PS-ring, MA and lB(M) = {b ∈ B | bm = 0,m ∈ M} are PS-modules and Soc(lB(M)) M = 0. Using the alternative of right T-module as triple (X,Y )f with X ∈ Mod-A, Y ∈ Mod-B and f : YM →... 展开更多
关键词 formal triangular matrix ring PS-ring CESS-module.
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The Auslander-type condition of triangular matrix rings
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作者 HUANG ChongHui HUANG ZhaoYong 《Science China Mathematics》 SCIE 2012年第8期1647-1654,共8页
Let R be a left and right Noetherian ring and n, k be any non-negative integers. R is said to satisfy the Auslander-type condition Gn(k) if the right fiat dimension of the (i + 1)-th term in a minimal injective r... Let R be a left and right Noetherian ring and n, k be any non-negative integers. R is said to satisfy the Auslander-type condition Gn(k) if the right fiat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 ≤ i ≤ n - 1. In this paper, we prove that R is Gn(k) if and only if so is a lower triangular matrix ring of any degree t over R. 展开更多
关键词 Auslander-type condition triangular matrix rings fiat dimension minimal injective resolutions mlnlm^l A.t r^nlllti^n~
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PP and Semihereditary Rings of Formal Triangular Matrices
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作者 范维丽 《Northeastern Mathematical Journal》 CSCD 2008年第2期143-149,共7页
In this paper the sufficient and necessary conditions are given for a formal triangular matrix ring to be right PP, generalized right PP, or semihereditary, respectively.
关键词 formal triangular matrix ring right PP-ring semihereditary ring
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Extensions of McCoy Rings 被引量:8
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作者 应志领 陈建龙 雷震 《Northeastern Mathematical Journal》 CSCD 2008年第1期85-94,共10页
A ring R is said to be right McCoy if the equation f(x)g(x) = 0, where y(x) and g(x) are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f(x)s = 0. It is proven that no proper ... A ring R is said to be right McCoy if the equation f(x)g(x) = 0, where y(x) and g(x) are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f(x)s = 0. It is proven that no proper (triangular) matrix ring is one-sided McCoy. It is shown that for many polynomial extensions, a ring R is right Mccoy if and only if the polynomial extension over R is right Mccoy. 展开更多
关键词 matrix ring McCoy ring polynomial ring upper triangular matrix ring
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Morphic-幂等环
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作者 甘海美 郭勇华 《曲阜师范大学学报(自然科学版)》 CAS 2024年第4期59-63,共5页
首先给出了morphic-幂等环的定义,并证明了环的直积是左morphic-幂等的当且仅当它的每个分量都是左morphic-幂等的.其次,给出了形式三角矩阵环是左morphic-幂等环的一个必要条件.最后,研究了左morphic-幂等环的平凡扩张.
关键词 左morphic-幂等环 形式三角矩阵环 平凡扩张
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Extensions of strongly π-regular general rings
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作者 王周 陈建龙 《Journal of Southeast University(English Edition)》 EI CAS 2007年第2期309-312,共4页
The concept of the strongly π-regular general ring (with or without unity) is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- reg... The concept of the strongly π-regular general ring (with or without unity) is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n+1y = zx^n+1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean. 展开更多
关键词 strongly π-regular general ring strongly clean general ring upper triangular matrix general ring trivial extension
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Study of Modules over 3 × 3 Formal Triangular Matrix Rings 被引量:1
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作者 SHI Mei Hua 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2008年第3期671-681,共11页
In this paper we carry out a study of modules over a 3 × 3 formal triangular matrix ringГ=(T 0 0 M U 0 N×UM N V)where T, U, V are rings, M, N are U-T, V-U bimodules, respectively. Using the alternative ... In this paper we carry out a study of modules over a 3 × 3 formal triangular matrix ringГ=(T 0 0 M U 0 N×UM N V)where T, U, V are rings, M, N are U-T, V-U bimodules, respectively. Using the alternative description of left Г-module as quintuple (A, B, C; f, g) with A ∈ mod T, B ∈ mod U and C ∈ mod V, f : M ×T A →B ∈ mod U, g : N ×U B → C ∈ mod V, we shall characterize uniform, hollow and finitely embedded modules over F, respectively. Also the radical as well as the socle of r (A + B + C) is determined. 展开更多
关键词 triangular matrix ring uniform module hollow module RADICAL socle.
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形式三角矩阵环上的F-Gorenstein平坦模
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作者 刘亚楠 杨刚 《吉林大学学报(理学版)》 CAS 北大核心 2023年第5期1029-1036,共8页
设T=A 0 U B是形式三角矩阵环,其中A,B是环,U是(B,A)-双模.利用Hom函子和伴随同构等理论,刻画形式三角矩阵环T上的F-Gorenstein平坦模结构,并证明若BU的平坦维数有限,U A的平坦维数有限且对任意的余挠左A-模C,有U■AC是余挠左B-模,则左T... 设T=A 0 U B是形式三角矩阵环,其中A,B是环,U是(B,A)-双模.利用Hom函子和伴随同构等理论,刻画形式三角矩阵环T上的F-Gorenstein平坦模结构,并证明若BU的平坦维数有限,U A的平坦维数有限且对任意的余挠左A-模C,有U■AC是余挠左B-模,则左T-模M_(1)/M_(2)φ^(M)是F-Gorenstein平坦模当且仅当M_(1)是F-Gorenstein平坦左A-模,Cokerφ^(M)是F-Gorenstein平坦左B-模,且φ^(M):U■AM 1→M_(2)是单射. 展开更多
关键词 平坦模 余挠模 形式三角矩阵环 F-Gorenstein平坦模
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三角矩阵环上投射余可解的Gorenstein AC-平坦模
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作者 秦军霞 张翠萍 《兰州理工大学学报》 CAS 北大核心 2023年第5期167-172,共6页
引入投射余可解的Gorenstein AC-平坦模的概念(简记为PGACF-模),给出这类模的等价条件.设T=(A 0 U B)是三角矩阵环,其中A,B是环,U是B,A-双模.在一定条件下,证明了如果M=(M_(1) M_(2))_(φ^(M))是PGACF左T-模,那么M_(1)是PGACF左A-模,Co... 引入投射余可解的Gorenstein AC-平坦模的概念(简记为PGACF-模),给出这类模的等价条件.设T=(A 0 U B)是三角矩阵环,其中A,B是环,U是B,A-双模.在一定条件下,证明了如果M=(M_(1) M_(2))_(φ^(M))是PGACF左T-模,那么M_(1)是PGACF左A-模,Cokerφ^(M)是PGACF左B-模,且φ^(M)是单同态;若PGACF-模的类对扩张封闭,则上述结论反过来也成立. 展开更多
关键词 投射余可解的Gorenstein AC-平坦模 三角矩阵环 absolutely clean模
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形式三角矩阵半环的广义导子
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作者 付婉婷 《科技资讯》 2023年第16期221-225,共5页
半环理论是代数理论上研究的热点问题。近年来,越来越多的研究人员注意到了半环理论在数学及其他研究领域的运用也非常普遍,在这些其他学科中有着广泛的应用。研究了形式三角矩阵半环Tri(A,M,B)上的广义导子的定义和表示形式。给出了半... 半环理论是代数理论上研究的热点问题。近年来,越来越多的研究人员注意到了半环理论在数学及其他研究领域的运用也非常普遍,在这些其他学科中有着广泛的应用。研究了形式三角矩阵半环Tri(A,M,B)上的广义导子的定义和表示形式。给出了半环上双半模上的拟同态映射f的定义。证明了半环Tri(A,M,B)上的任意的一个广义导子可以由半环A,B上的广义导子和(A,B)-双半模M上的一个拟同态映射来表示。 展开更多
关键词 广义导子 半环 半模 形式三角矩阵半环
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Triangular Matrix Representations of Rings of Generalized Power Series 被引量:4
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作者 Zhong Kui LIU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2006年第4期989-998,共10页
Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generali... Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]]. 展开更多
关键词 Generalized triangular matrix representation Twisted generalized power series ring PWPring Triangulating dimension
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关于拟正则Armendariz环
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作者 宋雪梅 《金陵科技学院学报》 2023年第1期88-92,共5页
引入拟正则Armendariz环并研究其性质。证明弱Armendariz环是拟正则Armendariz环,直积∏i∈I R i是拟正则Armendariz环当且仅当每个环R i(i∈I)是拟正则Armendariz环,同时证明R是拟正则Armendariz环当且仅当上三角矩阵环T n(R)(n≥2)是... 引入拟正则Armendariz环并研究其性质。证明弱Armendariz环是拟正则Armendariz环,直积∏i∈I R i是拟正则Armendariz环当且仅当每个环R i(i∈I)是拟正则Armendariz环,同时证明R是拟正则Armendariz环当且仅当上三角矩阵环T n(R)(n≥2)是拟正则Armendariz环,并通过例子说明任意环R上的全矩阵环M n(R)(n≥2)不是拟正则Armendariz环。 展开更多
关键词 拟正则元 弱ARMENDARIZ环 拟正则Armendariz环 半交换环 直积 上三角矩阵环
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形式三角矩阵环的零因子图 被引量:4
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作者 徐承杰 易忠 郑英 《数学杂志》 CSCD 北大核心 2013年第5期891-901,共11页
本文研究了形式三角矩阵环的零因子结构与零因子图的问题.利用零因子的性质及交换环零因子图的有关结论及分类讨论的方法,获得了形式三角矩阵环的零因子图直径为2的充要条件,推广了有限交换环的零因子图的相关结果.
关键词 形式三角矩阵环 无扰模 零因子 零因子图 直径
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三级三角矩阵环上的模范畴 被引量:5
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作者 史美华 曾庆怡 《浙江大学学报(理学版)》 CAS CSCD 北大核心 2005年第5期481-484,489,共5页
给出了三级三角矩阵环Γ的定义,通过建立一个等价函子F,证明了三角矩阵代数Γ上的有限生成模范畴mod Γ与Γ£是等价的范畴.利用伴随同构定理,得到了与Γ£同构的范畴Γ(~£).
关键词 三角矩阵环 等价函子 范畴
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