Injective minimal modules

Minimal modules over a ring with infinite center are studied. A characterization of rings R with center C, admitting minimal modules with C-torsion, is obtained. Minimal injective modules over a commutative ring are described.Translated fromAlgebra i Logika, Vol. 33, No. 2, pp. 211-226, March-April, 1994.     

Injective and automorphism-invariant non-singular modules

Every automorphism-invariant non-singular right A-module is injective if and only if the factor ring of the ring A with respect to its right Goldie radical is a right strongly semiprime ring.     

Injective modules and semistar operations

We approach the problem of classifying injective modules over an integral domain, by considering the class of semistar Noetherian domains. When working with such domains, one has to focus on semistar ideals: as a consequence for modules, we restrict our study to the class of injective hulls of co-semistar modules, those in which the annihilator ideal of each nonzero element is semistar. We obtain a complete classification of this class, by describing its elements as injective hulls of uniquely determined direct sums of indecomposable injective modules; if moreover, we consider stable semistar operations, then we can further improve this result, obtaining a natural generalization of the classical Noetherian case. Our approach provides a unified treatment of results on injective modules over various kinds of domains obtained by Matlis, Cailleau, Beck, Fuchs and Kim–Kim–Park.     

Injective modules and torsion functors*

A commutative ring is said to have ITI with respect to an ideal 𝔞 if the 𝔞-torsion functor preserves injectivity of modules. Classes of rings with ITI or without ITI with respect to certain sets of ideals are identified. Behavior of ITI under formation of rings of fractions, tensor products, and idealization is studied. Applications to local cohomology over non-noetherian rings are given.     

Injective modules over formal matrix rings

We describe injective modules over formal matrix rings.     

Injective modules and linear growth of primary decompositions

The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal in a commutative Noetherian ring has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , there exists a minimal primary decomposition with for all . The generalization involves a finitely generated -module and several ideals; the short proof is based on the theory of injective -modules.

     

Lattice repleteness and some of its applications to topology

We deal with the general concept of lattice repleteness. Specifically, we systematize the study of several important special cases of repleteness, namely, realcompactness, α-completeness, N-compactness, and Borel-completeness; we apply our general results on repleteness to specific lattices in topological spaces, in particular, to analytic spaces; we utilize the concept of $\text{G}$_δ-closure to obtain necessary or sufficient conditions for repleteness (this portion of our work generalizes important theorems of Mrówka on Stone-?echcompactification, of Frolik on realcompact spaces, and of Wenjen on realcompact spaces); finally, we extend the measure representation material of Varadarajan and then we utilize the results to obtain further applications to repleteness.     

Injective Rings

The purpose of this article is to determine the injective objects in some complete categories of rings. All rings are assumed to have identities and it is assumed that the homomorphisms preserve these identities. We recall that an object Q in a category is called injective if for every diagram where A′ → A is a monomorphism, there is a map A → Q making the triangle commute. The zero ring belongs to all the categories discussed and it is easy to see that it is an injective object. For the categories of commutative rings, strongly regular and commutative regular rings we show that the zero ring is the only injective by using the fact that an injective object must be a retract of any extension. We include in this section the known results which characterize the injective rings and p-rings. The second part of the paper discusses injectivity with respect to regular monomorphisms. Some necessary categorical background is given and it is then shown that results analagous with those of the first section hold (including the known Boolean and p-ring cases). In an abelian category all monomorphisms are regular, so in the study of the injective objects, for example injective modules, there are not two separate cases.     

Injective choice functions

The present paper is concerned with a combinatorial question called the “marriage problem.”. A criterion will be proved for the existence of an injective choice function of families with at most finitely many infinite members and a generalization of a theorem of H. A. Jung and R. Rado. We give a new proof of a theorem of J. Folkman.     

Main Injective Rings

We refer to those injective modules that determine every left exact preradical and that we called main injective modules in a preceding article, and we consider left main injective rings, which as left modules are main injective modules. We prove some properties of these rings, and we characterize QF-rings as those rings which are left and right main injective.     

内射和平坦L-半格

研究内射和平坦L-半格的一些性质,证明L-半格P是内射的当且仅当函子HomL(-,P)把单同态变为满映射。同时得到投射的L-半格是平坦的。