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991.
Shane P. Redmond 《代数通讯》2013,41(7):2389-2401
For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ? ?2 × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented. 相似文献
992.
Mikhail A. Chebotar 《代数通讯》2013,41(12):4339-4344
Let R be a 2-torsion free commutative ring with identity, and δ a nonzero derivation of R such that R is δ-prime. Then Rδ is a prime Lie ring and any nonzero ideal of Rδ contains an ideal of the form Jδ where J is a nonzero δ-ideal of R. 相似文献
993.
In this paper, algebras are finite dimensional over an algebraically closed field k, and modules are k-finite dimensional left modules. We prove the stable equivalence conjecture for algebras stably equivalent to algebras A with the following conditions: basic, connected and selfinjective; rad3A = 0 but rad2A ¬ 0; and the separated quiver Q3 A of the quiver QA of A consisting of more than two connected components. 相似文献
994.
In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group. 相似文献
995.
A. Van Daele 《代数通讯》2013,41(6):2341-2386
A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups. 相似文献
996.
A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext1(G, M) = 0 (resp. Tor1(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given. 相似文献
997.
We investigate several model theoretic minimalities in the framework of modules over a given serial ring R. 相似文献
998.
In this article we introduce a certain family of graded modules associated to a given module. These modules provide a natural extension of the notion of the associated graded ring of an ideal. We will investigate their properties. In particular, we will try to extend Ree' theorem on the associated graded ring of an ideal generated by a regular sequence to this context. 相似文献
999.
Cheng-Kai Liu 《代数通讯》2013,41(7):2396-2404
Let R be a commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-semiprime (or D-prime). We investigate the structure of the ideals of D as Lie rings. As a consequence, we give a necessary and sufficient condition for the ideals of D to be semiprime (or prime, respectively) Lie rings. 相似文献
1000.
Hiroki Miyahara 《代数通讯》2013,41(2):406-430
We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module. 相似文献