Desmond Sheiham
If R is a triangular 2x2 matrix ring, the columns, P and Q, are projective left R-modules. We describe the universal localization of R which makes invertible an R-module morphism P --> Q, generalizing a theorem of A.Schofield. We also describe the universal localization of R-modules.
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On Jordan superderivations and Jordan super-biderivations of trivial extensions and triangular matrix rings
本文研究平凡扩张与三角矩阵环上的Jordan超导子和Jordan超双导子，证明在一定条件下上三角矩阵环的Jordan超导子必为导子，且平凡扩张的Jordan超双导子必为Jordan双导子。
In this paper we extend to triangular n-matrix rings and Lie n-multiplicative map a result about Lie multiplicative maps on triangular algebras due to Xiaofei Qi and Jinchuan Hou.
Lie and Jordan Isomorphisms of Algebras of Triangular Matrices over Associative Rings
研究结合环上三角矩阵代数的李同构与若尔当同构，描述其相关性质。
The main purpose of this paper is to show that every Jordan centralizer and every Jordan two-sided centralizer is a centralizer on triangular rings without assuming unity. As an application, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. Some other related results are also discussed.
Guided by the research line introduced by Martindale III in [1] on the study of the additivity of maps, this article aims establish condi- tions on triangular matrix rings in order that an map ' satisfying '(ab + ba) = '(a)b + a'(b) + '(b)a + b'(a) for all a; b in a triangular matrix ring becomes additive.
In this article we give various formulates for compute the number of all coninvolutions over the group of upper triangular matrix with entries into the ring of Gaussian integers module $p$ and the ring of Quaternions integers module $p$, with $p$ an odd prime number.
Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
We describe isomorphisms between strongly triangular matrix rings that were defined earlier in Berkenmeier et al. (2000) as ones having a complete set of triangulating idempotents, and we show that the so-called triangulating idempotents behave analogously to idempotents in semiperfect rings. This study yields also a way to compute theoretically the automorphism groups of such rings in terms of corresponding automorphism groups of certain subrings and bimodules involved in their structure.
Let $\mathcal A$ and $\mathcal B$ be unital rings and $\mathcal M$ be a $(\mathcal A, \mathcal B)$-bimodule, which is faithful as a left $\mathcal A$-module and also as a right $\mathcal B$-module. Let ${\mathcal U}={\rm Tri}(\mathcal A, \mathcal M, \mathcal B)$ be the associated triangular ring. It is shown that every additive generalized Jordan (triple) higher derivation on $\mathcal U$ is a generalized higher derivation.
Given a ring $T_n, n\geqslant 2$, of lower triangular $n\times n$ matrices with entries from an arbitrary field $F$, a complete description is performed of the orbits of free cyclic submodules of $^2T_n$, under the action of the general linear group $GL_2(T_n)$. It is given its total number, which is equal to the Bell number $B_n$, and there are shown their representatives.
$n$-cotorsion pairs over formal triangular matrix rings
针对形式三角矩阵环$Λ$，通过研究其上特殊类，从环$A$和$B$的左/右$n$-余挠对构造出$Λ$的左/右$n$-余挠对，并举例验证结果。
For an element $a \in R$, let $η(a)=\{e\in R\mid e^2=e\mbox{ and }a-e\in \mbox{nil}(R)\}$. The nil clean index of $R$, denoted by NinA$(R)$, is defined as Nin$(R)=\sup \{\mid η(a)\mid: a\in R\}$. In this article we have characterized formal triangular matrix ring $\begin{pmatrix}A & M0 & B\end{pmatrix}$ with nil clean index $4$.
On Centrally Essential Subrings of Formal Triangular Matrix Rings
本文描述了形式三角矩阵环的一个大子类以及矩阵环 M_3(R) 的一个子类中的中心本质环。
Commuting Jordan derivations on triangular rings are zero
本文证明三角环上的交换Jordan导子恒为零，并推广到满足特定条件的2-无挠半素环或满足条件(P)的环。
Let $T=\left( \begin{array}{cc} R & M 0 & S \end{array} \right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cortés-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $\mathbb{D}^b(T{\text-} Mod)$ restricts to a recollement of its subcategory $\mathbb{D}^b(T{\text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective di…
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,σ)$ be the skew triangular matrix ring over a local ring $R$ where $σ$ is an endomorphism of $R$. We show that $T_2(R,σ)$ is strongly clean if and only if for any $a\in 1+J(R), b\in J(R)$, $l_a-r_{σ(b)}: R\to R$ is surjective. Further, $T_3(R,σ)$ is strongly clean if $l_{a}-r_{σ(b)}, l_{a}-r_{σ^2(b)}$ and $l_{b}-r_{σ(a)}$ are surjective for any $a\in U(R),b\in J(R)$. The necessary condition for $T_3(R,σ)$ to be strongly clean is also obtained.
Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over $T$ using the corresponding ones over $A$ and $B$. We show that when $U$ is relative (weakly) compatible we are able to describe the structure of $G_C$-projective modules over $T$. As an application, we study when a morphism in $T$-Mod has a special $G_CP(T)$-precover and when the class $G_CP(…
A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R \big(\in comm(a)\big)$ such that $a-e\in U(R)$. Let $R$ be a uniquely bleached ring. We prove, in this note, that $R$ is uniquely clean if and only if $R$ is abelian, and $T_n(R)$ is uniquely strongly clean for all $n\geq 1$, if and only if $R$ is abelian, $T_n(R)$ is uniquely strongly clean for some $n\geq 1$. In the commutative case, the more explicit results are obtained. These also generalize the main theorems in [6] and [7], and provide many new class of such rings.
Let $\T$ be a $2$-torsion free triangular ring and let $φ:\T\rightarrow \T$ be an additive map. We prove that if $\A φ(\B)+φ(\B)\A=0$ whenever $\A,\B\in \T$ are such that $\A\B=\B\A=0$, then $φ$ is a centralizer. It is also shown that if $τ:\T\rightarrow \T$ is an additive map satisfying $\label{t2} X,Y\in \T, \quad XY=YX=0\Rightarrow X τ(Y)+δ(X)Y+Yδ(X)+τ(Y)X=0$, where $δ:\T\rightarrow \T $ is an additive map satisfies $X,Y\in \T, \quad XY=YX=0\Rightarrow X δ(Y)+δ(X)Y+Yδ(X)+δ(Y)X=0$, then $τ(\A)=d(\A)+\A τ(\textbf{1})$, where $d:\T\rightarrow \T$ is a derivation and $τ(\textbf{1})$ lies in the…
There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coeffic…
In this paper, we show that a map $δ$ over a triangular ring $\mathcal{T}$ satisfying $δ(ab+ba)=δ(a)b+a τ(b)+δ(b)a+bτ(a)$, for all $a,b\in \mathcal{T}$ and for some maps $τ$ over $\mathcal{T}$ satisfying $τ(ab+ba)=τ(a)b+a τ(b)+τ(b)a+bτ(a)$, is additive. Also, it is shown that a map $T$ on $\mathcal{T}$ satisfying $T(ab)=T(a)b=aT(b)$, for all $a,b\in \mathcal{T}$, is additive. Further, we establish that if a map $D$ over $\mathcal{T}$ satisfies $(m+n)D(ab)=2mD(a)b+2naD(b)$, for all $a,b\in \mathcal{T}$ and integers $m,n\geq 1$, then $D$ is additive.
Let $T=\biggl(\begin{matrix} A&0\\ U&B \end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $\biggl(\begin{matrix} M_1\\ M_2\end{matrix}\biggr)_{φ^M}$ is Ding projective if and only if $M_1$ and $M_2/{\rm im}(φ^M)$ are Ding projective and the morphism $φ^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a righ…
Power commuting and centralizing maps on the ring of strictly upper triangular matrices
本文研究了严格上三角矩阵环上满足特定交换和中心化条件的映射性质，揭示了2幂交换映射与中心化映射的结构特征。
We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen-Macaulay modules. Three concrete examples of finite-dimensional algebras with the same singularity category are discussed.
Jordan homomorphisms of triangular algebras over noncommutative algebras
本文将Benkovič关于交换环上三角矩阵代数Jordan同态的刻画推广到非交换环，并去除加法挠假设，证明了T_n(R)的Jordan同态限制在子代数T_n^0(R)上必为标准形式。
Jordan Derivation On Generalized Triangular Matrix Rings
本文证明了广义三角矩阵环上的任意Jordan导子均为导子，推广了Brešar等人的已知结果。
Let $R$ be a left and right Noetherian ring and $n,k$ any non-negative integers. $R$ is said to satisfy the Auslander-type condition $G_n(k)$ if the right flat dimension of the $(i+1)$-st term in a minimal injective resolution of $R_R$ is at most $i+k$ for any $0 \leq i \leq n-1$. In this paper, we prove that $R$ is $G_n(k)$ if and only if so is a lower triangular matrix ring of any degree $t$ over $R$.
Bilinear maps on the ring of strictly upper triangular matrices
该研究在严格上三角矩阵环的背景下，推广了满足交换条件的线性映射的已有结论，完整刻画了满足对所有矩阵$X$都有$[f(X,X),X]=0$的双线性映射的结构。
Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane…
Formal Matrix Rings with the Involution Property
本文研究正式矩阵环的对合性质，通过Morita上下文构造并描述三角环中的对合，证明若所有对角环具此性质则任意有限阶正式矩阵环也具此性质，并给出例证。
Some Results on Triangular Coefficient Matrix Rings
本文引入三角系数矩阵环概念，研究其理想结构，并刻画 Hurwitz 多项式商环的根及基环与子环间的性质传递。
Given a block triangular matrix $M$ over a noncommutative ring with invertible diagonal blocks, this work gives two new representations of its inverse $M^{-1}$. Each block element of $M^{-1}$ is explicitly expressed via a quasideterminant of a submatrix of $M$ with the block Hessenberg type. Accordingly another representation for each inverse block is attained, which is in terms of recurrence relationship with multiple terms among blocks of $M^{-1}$. The latter result allows us to perform an off-diagonal rectangular perturbation analysis for the inverse calculation of $M$. An example is given …
A map $ϕ$ on an associative ring is called a multiplicative Lie derivation if $ϕ([x,y])=[ϕ(x),y]+[x,ϕ(y)]$ holds for any elements $x,y$, where $[x,y]=xy-yx$ is the Lie product. In the paper, we discuss the multiplicative Lie derivations on the triangular 3-matrix rings $\mathcal T={\mathcal T}_3(\mathcal R_i; \mathcal M_{ij})$. Under the standard assumption $Q_i\mathcal Z(\mathcal T)Q_i=\mathcal Z(Q_i\mathcal T Q_i)$, $i=1,2,3$, we show that every multiplicative Lie derivation $φ:\mathcal T\to\mathcal T$ has the standard form $φ=δ+γ$ with $δ$ a derivation and $γ$ a center valued map vanishing …
We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.
This paper will briefly survey some recent methods of localization in group rings, which work in more general contexts than the classical Ore localization. In particular the Cohn localization using matrices will be described, but other methods will also be considered.
General theory of localizations of rings and modules
本文旨在建立环与模的局部化/反演的最一般理论，引入并研究了若干新概念。
A ring $R$ is said to be right McCoy if the equation $f(x)g(x)=0,$ where $f(x)$ and $g(x)$ are nonzero polynomials of $R[x],$ implies that there exists nonzero $s \in R$ such that $f(x)s = 0$. It is proven that no proper (triangular) matrix ring is one-sided McCoy. If there exists the classical right quotient ring $Q$ of a ring $R$, then $R$ is right McCoy if and only if $Q$ is right 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. Other basic extensions of right McCoy rings are also studied.\left…
An element $a$ of a ring $R$ is called perfectly clean if there exists an idempotent $e\in comm^2(a)$ such that $a-e\in U(R)$. A ring $R$ is perfectly clean in case every element in $R$ is perfectly clean. In this paper, we investigate conditions on a local ring $R$ that imply that $2\times 2$ matrix rings and triangular matrix rings are perfectly clean. We shall show that for these rings perfect cleanness and strong cleanness coincide with each other, and enhance many known results. We also obtain several criteria for such a triangular matrix ring to be perfectly $J$-clean. For instance, it i…
It is proved that the localization of an injective module E, over a valuation ring R, at a prime ideal J, is injective if J is not the subset of zero-divisors of R or if J or E is flat. It follows that localizations of injective modules over h-local Prüfer domains are injective too.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prüfer domain of finite character, localizations of injective $R$-modules are injective.
We study automorphism groups of formal matrix algebras. We also consider automorphisms of ordinary matrix algebras (in particular, triangular matrix algebras).
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prüfer domain of finite character, localizations of injective $R$-modules are injective.
The concepts of localizable set, localization of a ring and a module at a localizable set are introduced and studied. Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a localizable set is a generalization of localization of a ring/module at a denominator set. For a semiprime left Goldie ring, it is proven that the set of maximal left localizable sets that contain all regular elements is equal to the set of maximal left denominator sets (and they are explicitly described). For a semiprime Goldie ring, it is proven that the following …
A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are rings and M is an S-R-bimodule. In the main theorem of this paper we show that if T is a tilting S-module, then under certain homological conditions on M as an S-module, one can extend T to a tilting complex over A inducing a derived equivalence between A and another triangular matrix ring specified by (S',R,M'), where the ring S' and the R-S'-bimodule M' depend only on T and M, and S' is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebr…
In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.
We extend to bimodules Schelter's localization of a ring with respect to Gabriel filters of left and right ideals. Our two-sided localization of bimodules provides an endofunctor on a convenient bicategory of rings with filters of ideals. This is used to study the Picard group of a ring relative to a filter of ideals.
In this paper we prove that if $R$ is a commutative refinement ring and $M$, $N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal $m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every regular matrix over $\frac{R}{J(R)}$ admits a diagonal reduction iff every regular matrix over $R$ admits a diagonal reduction.
In this article we study the behavior of left QI-rings under perfect localizations. We show that a perfect localization of a left QI-ring is a left QI-ring. We prove that Boyle's conjecture is true for left QI-rings with finite Gabriel dimension such that the hereditary torsion theory generated by semisimple modules is perfect. As corollary we get that Boyle's conjecture is true for left QI-rings which satisfy the restricted left socle condition, this result was proved first by C. Faith in \cite{faithhereditary}.
We consider wreath product decompositions for semigroups of triangular matrices. We exhibit an explicit wreath product decomposition for the semigroup of all n-by-n upper triangular matrices over a given field k, in terms of aperiodic semigroups and affine groups over k. In the case that k is finite this decomposition is optimal, in the sense that the number of group terms is equal to the group complexity of the semigroup. We also obtain some decompositions for semigroups of triangular matrices over more general rings and semirings.
The minimal primes of localizations of rings
本文研究环的局部化的极小素理想，在若干自然假设下给出其描述，并应用于半素环的最大商环，同时给出映射到中心的判据。
Let $\mathcal{T}$ be a triangular algebra over a commutative ring $\mathcal{R}$ and $φ: \mathcal{T} \times \mathcal{T}\longrightarrow \mathcal{T}$ be an arbitrary Lie biderivation of $\mathcal{T}$. We will address the question of describing the form of $φ$ in the current work. It is shown that under certain mild assumptions, $φ$ is the sum of an inner biderivation and an extremal biderivation and a some central bilinear mapping. Our results is immediately applied to block upper triangular algebras and Hilbert space nest algebras .
On Commuting graphs of triangular rings
本文研究了非交换环上$2\times 2$上三角矩阵环的交换图，计算了$R$为有限域时的边数、团数及独立数，并证明了当$R=\mathbb{Z}_{n}$非域时该图连通且直径为3。
This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix DGA's. An upper triangular matrix DGA has the form (R,S,M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R,S,M) and (S,M',R'), where the DG-bimodule M' is obtained from M and X and R' is the endomorphism DGA of a K-pr…
We explore elementary matrix reduction over certain rings characterized by their localizations. Let $R$ be a locally stable ring, we prove that $R$ is an elementary divisor ring if and only if $R$ is a Bezout ring. Elementary matrix reduction over some related localizations are also studied. Many known results on domains are thereby extended to general commutative rings which may contain many zero divisors.
The notion of a glider representation of a chain of normal subgroups of a group is defined by a new structure, i.e. a fragment for a suitable filtration on the group ring. This is a special case of general glider representations defined for a positively filtered ring $R$ with filtration $FR$ and subring $S = F_0R$. Nice examples appear for chains of groups, chains of Lie algebras, rings of differential operators on some variety or $V$-gliders for $W$ for algebraic varieties $V$ and $W$. This paper aims to develop a scheme theory for glider representations via the localizations of filtered modu…
In this paper, we introduce the notion of LSat, the left saturation closure of a subset of a module at a subset of the base ring, which generalizes multiple important concepts related to Ore localization. We show its significance in finding a saturated normal form for left Ore sets as well as in characterizing the units of a localized ring. Furthermore, LSat encompasses the notion of local closure of submodules and ideals from the realm of algebraic analysis, where it describes the result of extending a submodule or ideal from a ring to its localization and contracting it back again.
On the construction of Cohn's universal localization
本文研究了结合环上Cohn泛分式环的构造，指出其避免了Ore条件但依赖“伪Ore”条件，并证明该条件决定了构造中的等价关系。
Localization, Local--Global Transfer, and Hull Theory for $C4^{\ast}$-Modules over Commutative Rings
本文研究了C4-模、C4*模及其强C4*模的局部化与局部-全局转移理论，探讨了壳的结构及在交换环上的应用。
A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring $\mathbb Z_{(p)}[C_n]$ is clean, where $\mathbb Z_{(p)}$ is the localization of the integers at the prime $p$. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, $\mathcal O_K$ be its ring of integers, and $R$ be a localization of $\mathcal O_K$ at some prime ideal. We investigate when $R[G]$ is clean, where $G$ is a finite abelian group, a…
The unitary Cayley graph of upper triangular matrix rings
本文刻画了有限域上上三角矩阵环的酉 Cayley 图，证明其同构于完全图与 Hamming 图对极图的半强积，并计算了其直径、三径和团数等图论参数。
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