B-splines,Polytopes, and Their Characteristic D-modules

Given a polytope σ ? ? ^m , its characteristic distribution δ_σ generates a D-module which we call the characteristic D-module of σ and denote by M _σ. More generally, the characteristic distributions of a cell complex K with polyhedral cells generate a D-module M _K , which we call the characteristic D-module of the cell complex. We prove various basic properties of M _K , and show that under mild topological conditions on K, the D-module theoretic direct image of M _K coincides with the module generated by the B-splines associated to the cells of K (considered as distributions). We also give techniques for computing D-annihilator ideals of polytopes.     

The Galois representations associated to a Drinfeld module in special characteristic—I: Zariski density

Let ? be a Drinfeld A-module of rank r over a finitely generated field K. Assume that ? has special characteristic p₀ and consider any prime p≠p₀ of A. If End_K^sep(?)=A, we prove that the image of Gal(K^sep/K) in its representation on the p-adic Tate module of ? is Zariski dense in GL_r.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.     

Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢? _C  ? id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.     

On finitely generated module whose first nonzero fitting ideal is maximal

Let R be a Noetherian ring and M be a finitely generated R-module. Let I(M) be the first nonzero Fitting ideal of M. The main result of this paper asserts that when I(M) = Q is a regular maximal ideal of R, then M?R∕Q⊕P, for some projective R-module P of constant rank if and only if T(M)?QM. As a consequence, it is shown that if M is an Artinian R-module and I(M) = Q is a regular maximal ideal of R, then M?R∕Q.     

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = $\text{M}$_R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.     

Properties of a certain product of submodules

Let R be a commutative ring with identity, let M be an R-module, and let K ₁, . . . ,K _n be submodules of M: We construct an algebraic object called the product of K ₁, . . . ,K _n : This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M ⁿ   = M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M ⁿ is a projective (flat) R(M)-module.     

Serre Theorem for involutory Hopf algebras

We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel’d modules over H the dimension of which are invertible in k are Serre categories.     

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N?=?IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M)?=?{PM?|?P?∈?Spec(R) and P???M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.     

▵-Reductions of modules

Let △ be a multiplicatively closed set of finitely generated nonzero ideals of a ring R. Then the concept of a △ -reduction of an R -submodule D of an R -module A is introduced and several basic properties of such reductions are established. Among these are that a minimal △ -reduction B of D exists and that every minimal basis of B can be extended to a minimal basis of all R -submodules between B and D, when R is local and A is a finite R -module. Then, as an application, △ -reductions B of a submodule C with property (^?) are introduced, characterized, and shown to be quite plentiful. Here, (^?) means that (R ,M) is a local ring of altitude at least one, that △ = {M_n ; n ≥ 0} and that if D ? E are R -submodules between B and C, then every minimal basis of D can be extended to a minimal basis of E.     

Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N_R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M_P, as a vector space over the field R_P, is less than or equal to N_R. If a nonzero indecomposable R-module M is such that all the localizations M_P as vector spaces over the fields R_P have the same dimension r, for every minimal prime P of R, then r=1,2,3,4 or 6. Let n be an integer ≥8. We show that if M is an R-module such that the vector space dimensions of the M_P are between n and 2n−8, then M decomposes non-trivially. For each n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.     

Annihilators of power values of derivations in prime rings

In this paper we prove some results concerning annihilators of power values of derivations in prime rings. The following main theorem establishes a unified version of several earlier results in the literature:Let R be a prime ring with center Z and with extended centroid C,Q, its two-sided Martindale quotient ring, ρ a nonzero right ideal of R and D a nonzero derivation of R.Suppose that aD([x,y])ⁿ ∈ Z (D([x,y])ⁿa ∈ Z) for all x,y∈ρ where a∈Rand n is a fixed positive integer. If [ρ,ρ]ρ ≠ 0 and dim _C RC >4, then either aD(ρ) = 0 (a = 0 resp.) or D= ad(p) for some p∈ Qsuch that pρ = 0.     

The Brauer group of desingularization of moduli spaces of vector bundles over a curve

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M _C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧ ^r E = L. We show that the Brauer group of any desingularization of M _C (r; L) is trivial.     

On Isolated Submodules

Let R be a ring with identity and let M be a unital left R-module. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N ? K but L ? K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N ∩ IM = IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M.     

Notes on flatness and the quot functor on rings

For flat modules M over a ring A we study the similarities between the three statements,dim _k (P) ( _k (P)? _A M =dfor all prime ideals P of A, the Ap-module M _p is free of rank d for all prime ideals P of A, and M is a locally free J4-module of rank d. We have particularly emphasized the case when there is an>l-algebra B, essentially of finite type, and M is a finitely generated B-module.     

δ-M-Small and δ-Harada Modules

Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N? K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ _M -lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained.     

Prime non-Lie modules for Mal'tsev superalgebras

A module V for a superalgebra A is called prime if any two of its nonzero submodules have a nonzero intersection, and no nonzero submodule is annihilated by a nonzero ideal of A. We prove that if V is a prime module for a Mal'tsev superalgebra M = M₀+M₁, one of the following cases is realized:

Intersection multiplicities over Gorenstein rings

Let R be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for which this equality fails to hold. This then provides an example of a nonzero Todd class , and of a bounded free complex whose local Chern character does not vanish on this class. Received: February 11, 1999     

Submodules of Direct Sums of Compressible Modules

Let R be a ring. A right R-module M is called “essentially compressible” if it embeds in each of its essential submodules. Also a module X _R is called “completely essentially compressible” if every submodule of X _R is an essentially compressible R-module. In this aricle, it is shown that a right R-module M embeds in a direct sum of compressible right R-modules if and only if M _R is essentially compressible and every nonzero essentially compressible submodule of M _R contains a compressible submodule. Every essentially compressible R-module is shown to be retractable. Moreover, if either R _R has Krull dimension, or R is Morita equivalent to a right duo ring, then a right R-module embeds in a direct sum of compressible right R-modules if and only if it is completely essentially compressible.