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1.
Mamoru Kutami 《Applied Categorical Structures》2008,16(1-2):183-194
The notion of weak comparability was first introduced by K.C. O’Meara, to prove that directly finite simple regular rings
satisfying weak comparability must be unit-regular. In this paper, we shall treat (non-necessarily simple) regular rings satisfying
weak comparability and give some interesting results. We first show that directly finite regular rings satisfying weak comparability
are stably finite. Using the result above, we investigate the strict cancellation property and the strict unperforation property
for regular rings satisfying weak comparability, and we show that these rings have the strict unperforation property, which
means that nA ≺ nB implies A ≺ B for any finitely generated projective modules A, B and any positive integer n.
相似文献
2.
We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O'Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and -algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.
3.
Edgar E. Enochs Overtoun M. G. Jenda Jinzhong Xu 《Algebras and Representation Theory》1999,2(3):259-268
Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslander's result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslander's theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander's theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander's sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander's theorem when R is Gorenstein. 相似文献
4.
In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module. 相似文献
5.
Chaoling Huang 《印度理论与应用数学杂志》2011,42(5):261-277
In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is
closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category
of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R. 相似文献
6.
M.J. Dunwoody 《Journal of Pure and Applied Algebra》1980,16(3):249-265
The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K0(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K0(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension. 相似文献
7.
Mamoru Kutami 《代数通讯》2013,41(5):1579-1593
We study regular rings satisfying generalized almost comparability. First, we determine the forms of regular rings satisfying generalized almost comparability. Next, using the above result, we treat the strict cancellation property and the strict unperforation property for regular rings satisfying generalized almost comparability. 相似文献
8.
Li Huishi 《数学年刊B辑(英文版)》1994,15(4):463-468
It is proved that for a left Noetherian z-graded ring A,if every finitely generated graded A-module has finite projective dimension(i.e-,A is gr-regular)then every finitely generated A-module has finite projective dimension(i.e.,A is regular).Some applications of this result to filtered rings and some classical cases are also given. 相似文献
9.
LI HUISHI 《数学年刊B辑(英文版)》1994,(4)
NOETHERIANGT-REGULARRINGSAREREGULAR¥LIHUISHI(DepartmentofMathematics,ShaanxiNormalUnivisityXi'an710062,China.)Abstract:Itispr... 相似文献
10.
Peter Kálnai 《代数通讯》2019,47(1):88-100
We (re)introduce four ideal-related generalizations of classic module-theoretic notions: the ideal-superfluity, projective ideal-covers, the ideal-projectivity, and ideal-supplements. For a superfluous ideal I, the main theorem asserts the equivalence between the conditions: “I-supplements are direct summands in finitely generated projective modules”; “finitely generated I-projective modules are projective”; “projective modules with finitely generated factors modulo I are finitely generated”; “finitely generated flat modules with projective factors modulo I are projective.” Moreover, we provide a property of the ideal I which is sufficient for the equivalence to hold true. The property is expressed in terms of idempotent-lifting in matrix rings. 相似文献
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In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P <∞(Λ) if and only if P <∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]). 相似文献
14.
We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. IfH is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality. 相似文献
15.
Wolfgang Rump 《代数通讯》2013,41(9):3283-3299
ABSTRACT In this article, we study finitely generated reflexive modules over coherent GCD-domains and finitely generated projective modules over polynomial rings. In particular, we give a sufficient condition for a finitely generated reflexive module over a coherent GCD-domain to be a free module. By use of this result, we prove that every finitely generated projective R + [X]-module can be extended from R if R is a commutative ring with gl.dim(R) ≤ 2. 相似文献
16.
本文得到了相关比较Exchange环上模的替代性,并研究了幂比较Exchange 环上模的结构,进而提供了一类新的相关比较:Exchange环. 相似文献
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18.
Joseph Gubeladze 《Proceedings of the American Mathematical Society》1999,127(12):3493-3494
There are many affine subalgebras of polynomial rings with highly non-trivial projective modules, whose initial algebras (toric degenerations) are still finitely generated and have all projective modules free.
19.
The Comparable Structure of Exchange Rings 总被引:1,自引:0,他引:1
We obtain a new substitution for modules over exchange rings satisfying related comparability. Also we investigate the structure
of modules over exchange rings satisfying power comparability and provide a new class of exchange rings satisfying related
comparability.
Received June 1, 1998, Revised November 15, 1999, Accepted June 1, 2000 相似文献