Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

关于剩余类环的整体维数

If R/I is a flat R-module, the necessary-sufficient condition is given. Mo-reover, it is showh that the following two relations hold if a ring R has nonon-zero nilpotent one-sided ideals:( 1 ) LGD(R/Soc(R))≤LGD(R)≤LGD(R/Soc(R))+l ;(2) WD(R/Soc(R)) = WD (R).Particularly, the above relations holds for primitive rings and Von-Neumann regular rings .     

The Global Dimension of Rings of Differential Operators on Projective Spaces

We compute the global dimension of the twisted rings of globaldifferential operators D on complex projective n-space. As varies through the complex numbers, this global dimension takeson the values n, 2n and infinity. We also study the embeddingof D in the Weyl algebra A_n in the cases when the global dimensionis infinite. In these cases, we see that the flat dimensionof A_n as a D-module is infinite as well.     

The Structure of Schur Rings Over Cyclic Groups of Square-Free Order

We give the complete classification of Schur rings over cyclic groups of square-free order. In this classification we use the topological language originally suggested by Ya. Yu. Golfand. Each Schur ring over Z_n is uniquely determined by a finite topology L on the set of prime divisors of n and by a family {G _P }P L of finite groups satisfying an additional condition.     

On Triangulating Dimension of Rings

For any right essential overring T of a right FI-extending ring R, it is shown that 𝒯 dim(T) ≤ 𝒯dim(R), where 𝒯dim(?) is triangulating dimension of a ring. As a consequence, we show that for a ring R the maximal right ring of quotients, Q(R), is a direct product of finitely many prime rings if and only if Q(R) is semiprime and 𝒯dim(Q(R)) is finite. Some examples which illustrate and delimit the result are provided.     

Gabriel Dimension for Graded Rings

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕ _σ∈G R _σ with finite support if R _e has Gabriel dimension then has Gabriel dimension. Moreover, adding some lattice results, we prove that if has Gabriel dimension then also has Gabriel dimension. This research has been supported by grant MTM2005-3227 from MEC.     

Transitive Edge Coloring of Graphs and Dimension of Lattices

We explore properties of edge colorings of graphs dened by set intersections. An edge coloring of a graph G with vertex set V ={1,2, . . . , n} is called transitive if one can associate sets F ₁,F ₂, . . .,F _n to vertices of G so that for any two edges ij,kl E(G), the color of ij and kl is the same if and only if F _i F _j = F _k F _l . The term transitive refers to a natural partial order on the color set of these colorings.We prove a canonical Ramsey type result for transitive colorings of complete graphs which is equivalent to a stronger form of a conjecture of A. Sali on hypergraphs. This—through the reduction of Sali—shows that the dimension of n-element lattices is o(n) as conjectured by Füredi and Kahn.The proof relies on concepts and results which seem to have independent interest. One of them is a generalization of the induced matching lemma of Ruzsa and Szemerédi for transitive colorings. * Research supported in part by OTKA Grant T029074.     

自同态与环的弱维数

利用模的自同态研究ｖｏｎ Ｎｅｕｍａｎｎ正则环与环的弱维数,给出了ｖｏｎ Ｎｅｕｍａｎｎ正则环的新的刻划,同时也刻划了弱维数≤ｎ（ｎ≥０）的环。     

Commutative Schur Rings of Maximal Dimension

A commutative Schur ring over a finite group G has dimension at most s _G  = d ₁ + … +d _r , where the d _i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 ⁿ ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.     

Gr-凝聚环的分次FP-内射维数

引进了分次FP-内射维数，对Gr-凝聚环的分次FP-内射维数作了刻划，将Stentrom等人的若干工作推广到分次环上．     

非交换凝聚环上的FP-自内射维数

本文引进了W~(n)-模,对具有有限FP-自内射维数的非交换凝聚环作了刻划.所得结果推广了Stentr(?)m和Bass等人的工作.最后给出了这些结果在扩张闭模范畴中的应用.     

On Certain Schur Rings of Dimension 4

In this paper, we consider Schur rings on a finite group G of ordern(n-1) suchthat G has a partition with . Then Gis characterized as follows. (a) G has subgroups E andH of order n andn-1 respectively, and , or(b)G has subgroupsK andH( K) of order 2(n-1) and n-1 respectively,and . In addition assume that G has a subsetR of sizen-1 satisfying in the groupalgebraC[G]. Then G is characterized as a collineation groupof a projective plane of order n such that G has five orbits ofpoints of lengthsn(n-1), n, n-1, 1 and 1. In particular, we characterize projective planesof ordern admitting a quasiregular collineation group of order n(n-1)as the case that E and H are normal subgroups ofG.     

Square-Free Non-Cayley Numbers. On Vertex-Transitive Non-Cayley Graphs of Square-Free Order

A complete classification is given of finite primitive permutation groups which contain a regular subgroup of square-free order. Then a collection of square-free numbers n is obtained such that there exists a vertex-primitive non-Cayley graph on n vertices if and only if n is a member of .     

Witt Rings of Global Fields

Abstract

In 1989 DeMeyer et al. (DeMeyer,F.,Harrison,D.,Miranda,R. (1989). Quadratic forms over ? and Galois extensions of commutative rings. Mem. Amer. Math. Soc. 77(394)) described the Witt ring of the rational number field in terms of Hilbert symbols. The same has been done by the first author for global function fields. The aim of this paper is to obtain a similar result for any global field.     

Weak Dimension of FP-injective Modules Over Chain Rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3.     

Global Theory of Lattice-Finite Noetherian Rings

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal. Presented by H. Tachikawa     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

On the Krull Dimension of Rings of Transfer Functions

In this article, we prove that the Krull dimension of several commonly used classes of transfer functions of infinite dimensional linear control systems is infinite. On the other hand, we also show that the weak Krull dimension of the Hardy algebra , the disk algebra and the Wiener algebra is equal to 1. A. Sasane is supported by the Nuffield Grant NAL/32420.