Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f(X)∈R[X] and m(X)∈M[X], f(X)m(X) = 0 implies there exists a nonzero r∈R (resp., m∈M) with rm(X) = 0 (resp., f(X)m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.     

On right S-Noetherian rings and S-Noetherian modules

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.     

Secondary representation of some modules over a commutative ring

Let R be a commutative ring and let M be an R-module with the property that its zero submodule has a primary decomposition. Let E be an injective R-module with W.Ass _R (E) = Ass _R (E) (here W.Ass _R (E) denotes the set of weakly associated primes of E). Then we will show that Hom _R (M,E) has a secondary representation and we will specify the set of its attached prime ideals. This revised version was published online in June 2006 with corrections to the Cover Date.     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

The isomorphism problem for uniserial modules over an arbitrary ring

Abstract

First, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.     

Some remarks on growth of algebras over commutative noetherian rings

Using a growth function,GK defined for algebras over integral domains, we construct a generalization of Gelfand Kirillov dimensionGGK. GGK coincides with the classical no-tion of GK for algebras over a field, but is defined for algebras over arbitrary commutative rings. It is proved that GGK exceeds the Krull dimension for affine Noetherian PI algebras. The main result is that algebras of GGK at most one are PI for a large class of commutative Noetherian base rings including the ring of integers, Z. This extends the well-known result of Small, Stafford, and Warfield found in [11].     

Flatness and injectivity of simple modules over a commutative ring

In this short paper, the homological functors are used to show that a simple module is flat if and only if it is injective. It is used to give a characterization of Von Neumann regular ring, that is, a commutative ring is Von Neuman regular if and only if every simple module is injective.     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

Forcing linearity numbers of semicyclic modules over commutative Noetherian rings

In this paper we determine the forcing linearity numbers for semicyclic modules over commutative Noetherian rings.     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.