Weakly Stable Modules

ABSTRACT

A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ? R ⊕ C implies B ? C.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

Cubulating small cancellation groups

We study the B(6) and B(4)-T(4) small cancellation groups. These classes include the usual C(1/6) and C(1/4)-T(4) metric small cancellation groups. We show that every finitely presented B(4)-T(4) or word-hyperbolic B(6) group acts properly discontinuously and cocompactly on a CAT(0) cube complex. We show that finitely generated infinite B(6) and B(4)-T(4) groups have codimension 1 subgroups and thus do not have property (T). We show that a finitely presented B(6) group is wordhyperbolic if and only if it contains no subgroup.     

Simple-direct-modules

A right R-module M is called simple-direct-injective if, whenever, A and B are simple submodules of M with A?B, and B?^⊕M, then A?^⊕M. Dually, M is called simple-direct-projective if, whenever, A and B are submodules of M with M∕A?B?^⊕M and B simple, then A?^⊕M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ring R is uniserial (artinian serial) with J²(R) = 0 iff every simple-direct-projective right R-module is an SSP-module (SIP-module) iff every simple-direct-injective right R-module is an SIP-module (SSP-module).     

▵-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.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

Simple-Direct-Projective Modules

In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a right R-module M is called simple-direct-projective if, whenever A and B are submodules of M with B simple and M/A ? B ?^⊕M, then A ?^⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ring R is artinian and serial with J²(R) = 0 if and only if every simple-direct-projective right R-module is quasi-projective if and only if every simple-direct-projective right R -module is a D3-module. It is also shown that a ring R is uniserial with J²(R) = 0 if and only if every simple-direct-projective right R-module is a C3-module if and only if every simple-direct-injective right R -module is a D3-module.     

Utumi modules

A right R-module M is called a U-module if, whenever A and B are submodules of M with A?B and A ∩ B = 0, there exist two summands K and L of M such that A?^essK, B?^essL and K⊕L?^⊕M. The class of U-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely, the quasi-continuous, the square-free, and the automorphism-invariant modules. In this paper we show that the class of U-modules inherits some of the important features of the aforementioned classes of modules. For example, a U-module M is clean if and only if it has the finite exchange property, if and only if it has the full exchange property. As an immediate consequence, every strongly clean U-module has the substitution property and hence is Dedekind-finite. In particular, the endomorphism ring of a strongly clean U-module has stable range 1.     

Separative cancellation for projective modules over exchange rings

A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative. The research of the first and fourth authors was partially supported by a grant from the DGICYT (Spain) and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. That of the second author was partially supported by a grant from the NSF (USA). The final version of this paper was prepared while he was visiting the Centre de Recerca Matemàtica, Institut d'Estudis Catalans in Barcelona, and he thanks the CRM for its hospitality.     

The cancellable range of rings

We unify the cancellation property of rings with stable range one and the principal ideal domain by introducing a new notion which is called “cancellable range”. It is proved that if a ring R has cancellable range n for some positive integer n, then for any n-generated module B and any module implies B ≅ C; if R is a Noetherian ring and R has cancellable range n for any n ≧ 1, then R has the cancellation property. Received: 16 November 2004     

On the Howson Property of HNN-Extensions with Nilpotent Base Group and Amalgamated Free Products of Nilpotent Groups

We give necessary and sufficient conditions under which an amalgamated free product of finitely generated nilpotent groups is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated). Also we prove that if G = ? t, K | t ^?1 At = B ?, where K is a finitely generated and infinite nilpotent group and A, B non-trivial infinite proper subgroups of K, then G is not a Howson group. The problem of deciding when an ascending HNN-extension of a finitely generated nilpotent group is a Howson group is still open.     

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.     

On algebraic properties of monotone clones

Let R ₈ denote the 8-element bounded tower. G. Tardos has shown that C(R ₈), the clone of all monotone functions on R ₈, is not finitely generated. In this paper we show that the clone of all nonsurjective functions is finitely generated.     

Von Neumann Regular Rings Satisfying Weak Comparability

The notion of weak comparability was first introduced by K.C. O’Meara, to prove that directly finite simple regular rings satisfying weak comparability must be unit-regular. In this paper, we shall treat (non-necessarily simple) regular rings satisfying weak comparability and give some interesting results. We first show that directly finite regular rings satisfying weak comparability are stably finite. Using the result above, we investigate the strict cancellation property and the strict unperforation property for regular rings satisfying weak comparability, and we show that these rings have the strict unperforation property, which means that nA ≺ nB implies A ≺ B for any finitely generated projective modules A, B and any positive integer n.      

On Modules with Few Minimax Cocentralizers

Let R be a ring and G a group. An R-module A is said to be minimax if A includes a noetherian submodule B such that A/B is artinian. The authors study a ?G-module A such that A/C _A (H) is minimax (as a ?-module) for every proper not finitely generated subgroup H.     

On Rings for Which Finitely Generated Ideals Have Only Finitely Many Minimal Components

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R are always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over R as well as to R-algebras which are finitely presented as R-modules.     

Constants of Derivation and Differential Ideals

ABSTRACT

Let R be a differential domain finitely generated over a differential field F of characteristic 0. Let C be the subfield of differential constants of F. This paper investigates conditions on differential ideals of R that are necessary or sufficient to guarantee that C is also the set of constants of differentiation of the quotient field, E, of R. In particular, when C is algebraically closed and R has a finite number of height one differential prime ideals, there are no new constants in E. An example where F is infinitely generated over C shows the converse is false. If F is finitely generated over C and R is a polynomial ring over F, sufficient conditions on F are given so that no new constants in E does imply only finitely many height one prime differential ideals in R. In particular, F can be Q¯(T) where T is a finite transcendence set.