On multiplicities of simple subquotients in generalized Verma modules

We reduce the problem on multiplicities of simple subquotients in an -stratified generalized Verma module to the analogous problem for classical Verma modules.     

Extending modules which are direct sums of injective modules and semisimple modules

It is proved that, if R is a right Noetherian ringM ₁ is an injective right R-module and M ₂ is a semisimple right R-module, then the right R-module M ₁ + M ₂ is extending if and only if M ₂ is (M ₁/Soc(M ₁))-injective.     

Single elements and module isomorphisms of some operator algebra modules

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the Alg-module is initiated, where is a completely distributive subspace lattice on a Hilbert space . Furthermore, as an application of single elements, we study module isomorphisms between norm closed Alg-modules, where is a nest, and obtain the following result: Suppose that are norm closed Alg-modules and that is a module isomorphism. Then and there exists a non-zero complex number such that .

     

I-提升模的有限直和

I-提升模的直和不一定是I-提升模.本文给出了使I-提升模的直和仍是I-提升模成立的条件,即证明了当M=M1⊕M2,其中M1和M2是I-提升的.如果Mi是Mj-投射的（i,j=1,2）或M是duo模,则M是I-提升的.     

Decompositions of modules lacking zero sums

A module over a semiring lacks zero sums (LZS) if it has the property that v +w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, this property always holds for modules over an idempotent semiring and related semirings, so arises for example in tropical mathematics.A direct sum decomposition theory is developed for direct summands (and complements) of LZS modules: The direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated “strongly projective” module is a finite direct sum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonal primitive idempotents of R). This leads to an analog of the socle of “upper bound” modules. Some of the results are presented more generally for weak complements and semi-complements. We conclude by examining the obstruction to the “upper bound” property in this context.     

Embeddings of canonical modules and resolutions of connected sums

For an ideal $I_{m,n}$ generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by $I_{m,n}$. Using this embedding, we give a resolution of connected sums of several copies of certain Artin $\mathsf{k}$-algebras where $\mathsf{k}$ is a field.     

Rings characterized by direct sums of cs modules

It is shown that a ring for which every CS right module is ∑CS is right artinian. As a consequence, it is also shown that over a ring R every direct sum of CS right R-modules is CS iff R is right artinian and the composition length of every uniform right R-module is at most 2.     

Design isomorphisms and group isomorphisms

In this paper conditions are given which imply that a design isomorphism between designs with group actions is in fact a group isomorphism. The conditions are geometric. Automorphism groups are then calculated for a family of designs using the geometric conditions.The research in the paper was conducted at Loyola University of Chicago.     

Endomorphisms of free modules as sums of four quadratic endomorphisms

We will prove that every endomorphism of a free right R-module of infinite rank can be decomposed as a sum of four endomorphisms which satisfy some fixed polynomial identities.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Projective covers as subquotients of enlargements

We show that not only completions of systems in various senses but also projective covers in the category of compact Hausdorff spaces may be obtained as subquotients of enlargements.