The Stable Class of the Augmentation Ideal

In [F.E.A. Johnson, Stable Modules and the D(2)-Problem, LMS Lecture Notes In Mathematics, vol. 301, CUP (2003)], for finite groups G, we gave a parametrization of the stable class of the augmentation ideal of Z[G] in terms of stably free modules. Whilst the details of this parametrization break down immediately for infinite groups, nevertheless one may hope to find parallel arguments for restricted classes of infinite groups. Subject to the restriction that Ext¹(Z, Z[G]) = 0, we parametrize the minimal level in Ω₁(Z) by means of stably free modules and give a lower estimate for the size of Ω₁(Z).     

On Flatness and Completion for Infinitely Generated Modules over Noetherian Rings

Let A be a noetherian commutative ring, and let 𝔞 be an ideal in A. We study questions of flatness and 𝔞-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and 𝔞-adically free A-module.     

Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod _A (d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in mod _A (d) is nonsingular if and only if Ext _A ²(M,M)=0.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Weakly dual automorphism invariant and superfluous cotightness

Abstract

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules.     

ON THE FINITENESS PROPERTIES OF THE GENERALIZED LOCAL COHOMOLOGY MODULES

Abstract

The question, posed in Crawley and Jónsson (Crawley, P., Jónsson, B. (1964). Refinements for infinite direct decompositions of algebraic systems. Pacific J. Math. 14:797–855), whether the finite exchange property implies the unrestricted exchange property for any modules, is still open. In this note we obtain the equivalence of the finite exchange property and the unrestricted exchange property for the class of modules whose endomorphism rings are Abelian.     

Valuation Domains Whose Products of Free Modules Are Separable

It is proved that if R is a valuation domain with maximal ideal P and if R _L is countably generated for each prime ideal L, then R ^R is separable if and only R _J is maximal, where J = ∩ _n∈? P ⁿ .     

On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.     

Free Modules over Cartesian Closed Topological Categories

The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Gorenstein tilting pairs

Abstract

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.     

Another type of dimensions of modules and rings

Abstract

We say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension.     

Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).     

Prime virtually semisimple modules and rings

Abstract

We show that the multiplication algebra of a nondegenerate Jordan algebra is a semiprime algebra.     

Monomorphisms in the category of firm modules

Abstract

In this article, we consider the category of unitary right modules over an (associative) ring and the category of firm right modules over an idempotent ring. We study monomorphisms in these categories and give conditions under which morphisms are monomorphisms in the category of firm modules. We also prove that the lattice of categorically defined subobjects of a firm module is isomorphic to the lattice of unitary submodules of that module.     

Sequentially Cohen-Macaulay Modules Under Base Change

ABSTRACT

Assume that ?:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ? _R  S.     

On co-isosimple modules and co-isoradical of modules

Abstract

In this paper we study “co-isosimple” modules, that is, those modules which are isomorphic to all of its non-zero quotients modules. This allows us to define and study “isomaximal” submodules, “isomax” modules and the “co-isoradical” of a module. We study some of its basic properties and we give a characterization of left V-ring using these concepts.

Communicated by Alberto Facchini     

Inégalité de Bernstein Pour Une Localisation Simple de L'espace Symplectique Quantique

A Bernstein-type inequality is found for a simple localization of the coordinate ring of the quantized symplectic space 𝒪 _q (𝔰 𝔭 ^2×n ), where q is not a root of unity. Some holonomic modules and an analog of the Bernstein polynomial are also computed.

On trouve une inégalité de type “inégalité de Bernstein” pour une localisation simple de l'anneaux des coordonnées de l'espace symplectique quantique 𝒪 _q (𝔰 𝔭 ^2×n ), où q est non racine de l'unité. Des modules holonomes et un analogue du polynôme de Bernstein sont aussi calculés.