Manuel Saorin
Suppose that $A$ is a semiprimary ring satisfying one of the two conditions: 1) its Yoneda ring is generated in finite degrees; 2) its Loewy length is less or equal than three. We prove that the global dimension of $A$ is finite if, and only if, there is a $m>0$ such that $Ext_A^n(S,S)=0$, for all simple $A$-modules $S$ and all $n\geq m$.
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We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring $S\supseteq R$ and a set $X\subseteq S$ such that $R_0=\Cent_R(X):={r\in R \suchthat xr=rx \forall x\in X}$; in particular, centralizers of subsets of $R$ are semi-invariant subrings. We prove various properties of semi-invariant subrings and show how they can be used for various applications including: (1) The center of a semiprimary (resp. right perfect) ring …
Hopkins-Levitzki Type Theorems for Groupoid Graded Rings
该研究针对群胚分次环，引入分次理想的左/右逐对象幂零性概念，推广Hopkins-Levitzki定理，证明三类Γ₀-Artinian环为Γ₀-Noetherian环，并用反例说明相关结论。
In this paper we work with unital twisted partial actions. We investigate ring theoretic properties of partial crossed products as artinianity, noetherianity, perfect property, semilocalproperty, semiprimary property and we also study the Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products Also we study when the partial crossed products are Frobenius and symmetric algebras.
Radical preservation and the finitistic dimension
本文引入根保持概念与拟一致Loewy长度的有界箭图代数，证明相关同态可反映维数有限性，且这类代数及半准素环有有限大有限维数，还构造了现有结果无法解释的代数族。
Endochain conditions on modules and $p$-quasi-Frobenius rings
本文引入$p$-拟弗罗贝尼乌斯环等三类环，通过链对称定理及三类环的性质，解决了Faith-Menal、Faith、FGF猜想的主要类似问题，且以反例验证假设的必要性。
Cellular Noetherian algebras with finite global dimension are split quasi-hereditary
本文证明了有限整体维数的细胞诺特代数在正则交换诺特环上是分裂拟遗传的且结构唯一，给出了整体维数公式和有限维数上界，并应用于Schur代数。
Global Representation Ring and Knutson Index
本文将Knutson指数推广到一般交换环，研究其在Burnside环和全局表示环上的表现，并引入统一特征标表与Burnside标记表的全局表，探讨可恢复的群性质。
In this paper we study the finiteness of global Gorenstein AC-homological dimensions for rings, and answer the questions posed by Becerril, Mendoza, Pérez and Santiago. As an application, we show that any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category, which improves the known result by Beligiannis.
In this paper, the notion of strongly G_C-projective and injective modules is introduced, where C is a semidualizing module. Using these modules we can obtain a new characterization of G_C-projective and injective modules, similar to the one of projective modules by the free modules. We then define and study the global dimensions of rings relative to a semidualizing module C, and prove that the global G_C-projective dimension of a ring R is equal to the global G_C-injective dimension of R.
Given a partial (resp. a global) action $α$ of a connected finite groupoid $G$ on a ring $A$, we determine necessary and sufficient conditions for the partial (resp. global) skew groupoid ring $A\star_α G$ to be a separable extension of $A$.
The global dimension of a ring governs many useful abilities. For example, it is semi-simple if the global dimension is 0, hereditary if it is 1 and so on. We will calculate the global dimension of a Crystalline Graded Ring, as defined in the paper by E. Nauwelaerts and F. Van Oystaeyen, Introducing Crystalline Graded Algebras, Algebras and Representation Theory vol 11(2008), no. 2, 133--148.. We will apply this to derive a condition for the Crystalline Graded Ring to be semiprime. In the last section, we give a little bit of attention to the Krull-dimension.
Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension
本文证明具有有限 Gorenstein 弱整体维数的环满足同伦范畴的三角等价，并将其应用于 Ding-Chen 环和 Gorenstein n-相干环的 Grothendieck 对偶。
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of…
We expand on two existing characterizations of rings of Gorenstein (weak) global dimension zero and give two new characterizations of rings of finite Gorenstein (weak) global dimension. We also include the answer to a question of Y.~Xiang on Gorenstein weak global dimension of group rings.
In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introd…
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。
In this paper we describe how to give a particular global category of rings and modules the structure of a relaxed multi category, and we describe an algebra in this relaxed multi category such that vertex algebras appear as such algebras.
For a commutative local ring $R$, consider (noncommutative) $R$-algebras $Λ$ of the form $Λ= End_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $Λ$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of $Spec R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $C$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $Λ$ with finite global dimension and which is maximal Cohen--Macaulay…
Regular algebras of dimension four associated to coordinate rings of rank-two quadrics
本文分类整体维数四、具有四生成元多项式环 Hilbert 级数且满射到秩二二次曲面扭曲齐次坐标环的连通分次二次 AS-正则代数，补充了已有秩三、秩四情形。
We study Morita rings $Λ_{(ϕ,ψ)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$ in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $ϕ$ and $ψ$ are zero. Further we give bounds for the global dimension of a Morita ring $Λ_{(0,0)}$, regarded as an Artin algebra, in terms of the global dimensions of $A$ and $B$ in the case when both $ϕ$ and $ψ$ are zero. We illustrate our bounds with some examples. Finally we inve…
One-sided Gorenstein rings
本文证明Iwanaga--Gorenstein环的若干特征可推广至单侧情形，无需诺特假设，并在Gorenstein整体维数、模维数上确界及有限维数间建立新关系。
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units. This has been investigated for several types of rings related to global fields, most importantly rings of algebraic integers. We also state some open problems and conjectures which we consider to be important in this field.
We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 1, 2, or 3, and the global dimension of ko is 4 or 5.
The MacWilliams extension theorem is investigated for various weight functions over finite Frobenius rings. The problem is reformulated in terms of a local-global property for subgroups of the general linear group. Among other things, it is shown that the extension theorem holds true for poset weights if and only if the underlying poset is hierarchical. Specifically, the Rosenbloom-Tsfasman weight for vector codes satisfies the extension theorem, whereas the Niederreiter-Rosenbloom-Tsfasman weight for matrix codes does not. A short character-theoretic proof of the well-known MacWilliams extens…
In this paper, we investigate the weak Gorenstein global dimensions. We are mainly interested in studying the problem when the left and right weak Gorenstein global dimensions coincide. We first show, for GF-closed rings, that the left and right weak Gorenstein global dimensions are equal when they are finite. Then, we prove that the same equality holds for any two-sided coherent ring. We conclude the paper with some examples and a brief discussion of the scope and limits of our results.
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(…
In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.
Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimensi…
Ring Theory states that a ring is an algebraic structure where two binary operations can be performed among the elements addition and multiplication. Binarization is a method of image processing where values within pixels are reduced to a scale from zero to one, with zero representing the most absence of light and one representing the most presence of light. Currently, sonograms are implemented in scanning for congestive heart failure. However, the renowned Playboy Bunny symbol representing the ailment becomes increasingly difficult to isolate due to surrounding organs and lower quality image …
By analysing the structure of the associated graded ring with respect to certain filtrations, we deduce a number of good properties of iterated local skew power series rings over appropriate base rings. In particular, we calculate the Krull dimension, prime length and global dimension in well-behaved cases, obtaining lower bounds to complement the upper bounds obtained by Venjakob and Wang.
Homological invariant properties under cleft extensions
本文研究环的cleft扩张下Gorenstein弱整体维数的有限性不变性，并比较相对奇点范畴，应用于θ-扩张和Morita context环。
We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.
Finiteness criteria for Gorenstein homological dimension and some invariants of groups
本文研究了交换环上群的Gorenstein同调维数有限性准则，估计了群环的Gorenstein弱整体维数，证明了上同调维数对同调维数的界定，并比较了广义上同调与同调维数。
Properties of quasi-projective dimension over abelian categories
本文研究阿贝尔范畴中拟投射维数的基本性质，引入左Noether环的拟整体维数，并通过Nakayama代数实例展示其与整体维数的差异。
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