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1.
We show that a weak-injective module over an integral domain need not be pure-injective (Theorem 2.3). Equivalently, a torsion-free Enochs-cotorsion module over an integral domain is not necessarily pure-injective (Corollary 2.4). This solves a well-known open problem in the negative.In addition, we establish a close relation between flat covers and weak-injective envelopes of a module (Theorem 3.1). This yields a method of constructing weak-injective envelopes from flat covers (and vice versa). Similar relation exists between the Enochs-cotorsion envelopes and the weak dimension ?1 covers of modules (Theorem 3.2). 相似文献
2.
Edgar Enochs S. Estrada J. R. García-Rozas L. Oyonarte 《Algebras and Representation Theory》2004,7(4):441-456
The study of flat covers and cotorsion envelopes has turned out to be very useful since their existence was proved in [3] for the category of R-modules. The problem is even more interesting in categories of sheaves on a topological space, because these categories do not have enough projectives. But the existence of flat covers and cotorsion envelopes allow us to form flat and cotorsion resolutions to compute cohomology. 相似文献
3.
Alberto Facchini 《Israel Journal of Mathematics》1985,52(1-2):129-139
We describe the torsion-free covers of cyclic modules, the pure-injective envelopes of ideals, the maximal immediate extensions
of localizations and the injective envelopes of cyclics over valuation domains. We study the relations among these modules.
This paper generalizes some results of Banaschewski, Cheatham, Enochs and Nishi.
Supported by Ministero della Pubblica Istruzione and GNSAGA (CNR). 相似文献
4.
《代数通讯》2013,41(4):1453-1470
Abstract In this paper, we show that if R is a local Cohen–Macaulay ring admitting a dualizing module Ω, then Ω-Gorenstein projective and flat covers and Ω-Gorenstein injective envelopes exist for certain modules. These results generalize the well known results for local Gorenstein rings. 相似文献
5.
6.
Injective and flat covers,envelopes and resolvents 总被引:11,自引:0,他引:11
Edgar E. Enochs 《Israel Journal of Mathematics》1981,39(3):189-209
Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left noetherian. Flat envelopes are defined and shown to exist for all modules over a regular local ring of dimension
2. Using injective covers, minimal injective resolvents can be defined. 相似文献
7.
《Quaestiones Mathematicae》2013,36(1):35-44
We prove the existence of Gorenstein injective envelopes and covers over n-perfect rings for some classes of modules associated with a dualizing bimodule. 相似文献
8.
9.
Let R be a commutative ring. We investigate the relationship between (pre)covers ((pre)envelopes) of an R-module and the counterparts of the corresponding homomorphism module or tensor product module. Some applications are also given. 相似文献
10.
In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied. 相似文献
11.
We study classes of relative injective and projective comodules and extend well-known results about projective comodules given
in [7]. The existence of covers and envelopes by these classes of comodules is also studied and used to characterize the projective
dimension of a coalgebra. We also compare this homological coalgebra with the very intensively studied homological algebra
of the dual algebra (see [5]).
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
12.
It is known that tilting classes are of finite type, while cotilting classes are not always of cofinite type. We investigate this phenomenon. By using a bijection between definable classes of left modules and definable classes of right modules, we prove that it reflects the asymmetry existing between the notions of covers and envelopes or, otherwise stated, right and left approximations.In particular we show that there exist definable torsion classes containing the injective modules which are not tilting classes. 相似文献
13.
Lidia Angeleri Hügel Alberto Tonolo Jan Trlifaj 《Algebras and Representation Theory》2001,4(2):155-170
We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes. 相似文献
14.
Harold P. Benson 《Computational Optimization and Applications》2004,27(1):5-22
Convex and concave envelopes play important roles in various types of optimization problems. In this article, we present a result that gives general guidelines for constructing convex and concave envelopes of functions of two variables on bounded quadrilaterals. We show how one can use this result to construct convex and concave envelopes of bilinear and fractional functions on rectangles, parallelograms and trapezoids. Applications of these results to global optimization are indicated. 相似文献
15.
Claudia Chaio 《代数通讯》2013,41(12):4419-4424
We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of these maps, which is independent from the maximal length of the indecomposable modules. 相似文献
16.
We study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions. Specifically, we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach, due to McCormick. The standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently. We show that for a multilinear function having a single product term, this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero. We then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms. Finally, for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is always within a constant of the difference between the concave and convex envelopes. These results, along with numerical examples we provide, give insight into how to construct strong relaxations of multilinear functions. 相似文献
17.
A module M is called strongly FP-injective if Exti(P,M) = 0 for any finitely presented module P and all i≥1. (Pre)envelopes and (pre)covers by strongly FP-injective modules are studied. We also use these modules to characterize coherent rings. An example is given to show that (strongly) FP-injective (pre)covers may fail to be exist in general. We also give an example of a module that is FP-injective but not strongly FP-injective. 相似文献
18.
The subject of envelopes has been part of differential geometry from the beginning. This paper brings a modern perspective
to the classical problem of envelopes of families of affine hyperplanes. In the process, the classical results are generalized
and unified. 相似文献
19.
Mohit Tawarmalani Jean-Philippe P. Richard Chuanhui Xiong 《Mathematical Programming》2013,138(1-2):531-577
In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions. 相似文献
20.
John M. Ball Bernd Kirchheim Jan Kristensen 《Calculus of Variations and Partial Differential Equations》2000,11(4):333-359
We prove that the quasiconvex envelope of a differentiable function which satisfies natural growth conditions at infinity
is a function. Without the growth conditions the result fails in general. We also obtain results on higher regularity (in the
sense of ) and similar results for other types of envelopes, including polyconvex and rank-1 convex envelopes.
Received January 11, 2000/ Accepted January 14, 2000 / Published online June 28, 2000 相似文献