t-Extending Modules and t-Baer Modules

We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z ₂(R _R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

On GC2 modules and their endomorphism rings

Let R be a ring. A module M_R is said to be GC₂ if for any N≤ M with N? M, N is a direct summand of M. In this article, we give some characterizations and properties of GC₂ modules and their endomorphism rings, and many results on C ₂ modules and GC₂ rings are generalized to GC₂ modules.     

Modules Whose t-Closed Submodules Have a Summand as a Complement

We define and investigate T ₁₁-type modules as a generalization of t-extending modules, and modules satisfying C ₁₁ condition. A module M is said to be T ₁₁-type if every t-closed submodule of M has a complement which is a direct summand. Direct sums of T ₁₁-type modules inherit the property. Some equivalent conditions for a module M to be T ₁₁-type are given. We characterize a module M for which every direct summand satisfies T ₁₁ condition. If R _R is T ₁₁-type, then R/Z ₂(R _R ) is a C ₂ ring if and only if it is a von Neumann regular ring. Applying this result, we characterize a right t-extending (resp., finitely Σ-t-extending, or Σ-t-extending) ring R for which R/Z ₂(R _R ) is von Neumann regular.     

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A ₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl ₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.      

Irreducible weight modules for the Virasoro-like algebra and its q-analog

In this paper, using Larsson’s functor with irreducible 𝔰𝔩₂-modules V, we construct a class of ?²-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.     

On modules with <Emphasis Type="Italic">d</Emphasis>-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.     

Quasi-cotilting modules and hereditary quasi-cotilting modules

In this paper, we firstly give some basic properties on quasi-cotilting modules. With the help of these properties, we obtain a Quasi-cotilting Theorem (see Theorem 2.8). In particular, the Cotilting Theorem in [5 Colpi, R. R. (2000). Cotilting bimodules and their dualities. Lect. Notes Pure Appl. Math. 210:81–94. [Google Scholar]] is a corollary of our result. Finally, we introduce a new notion, hereditary quasi-cotilting modules, and mainly prove that the class consisting of all Δ-reflexive modules for a hereditary quasi-cotilting module is closed under submodules.     

Varieties and cohomology of infinitely generated modules

Suppose that k is a field of characteristic p and that G is a finite group having p-rank at least three. Given a regular sequence of length 2 in H*(G, k), we show how to construct an infinitely generated module whose support variety is strictly smaller than the small support of its cohomology. Received: 25 January 2008     

On multiplication and comultiplication modules

This article deals with some results concerning multiplication and comultiplication modules over a commutative ring.     

On a class of semicommutative modules

Let R be a ring with identity, M a right R-module and S = End _R (M). In this note, we introduce S-semicommutative, S-Baer, S-q.-Baer and S-p.q.-Baer modules. We study the relations between these classes of modules. Also we prove if M is an S-semicommutative module, then M is an S-p.q.-Baer module if and only if M[x] is an S[x]-p.q.-Baer module, M is an S-Baer module if and only if M[x] is an S[x]-Baer module, M is an S-q.-Baer module if and only if M[x] is an S[x]-q.-Baer module.     

Tensor product of C-injective modules

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness     

Cotilting modules are pure-injective

We prove that a cotilting module over an arbitrary ring is pure-injective.

     

Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties

We study the following question: when is the right adjoint of the forgetful functor from the category of -Doi-Hopf modules to the category of -modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that and the smash product are isomorphic as -bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case , and this leads to the notion of -Frobenius -module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if is finite dimensional and unimodular.

     

Complexity of degenerations of modules

A module M over an associative algebra A over an algebraically closed field k is said to degenerate to a module N if N belongs to the closure of the isomorphism class of M in the algebraic variety of d-dimensional A-modules, . We associate a non-negative integer to a degeneration , its complexity, and study its properties. Received: January 30, 2001     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk. This work is partially supported by the National Natural Science Foundation of China (Grant No. 10525106), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program