On the exactness of the completion functor

Torsion theories over commutative rings

Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.

In this paper, we study generalized Reed-Solomon codes (GRS codes) over commutative and noncommutative rings, we show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context, and we investigate their complexities. Under some hypothesis, the study of noncommutative GRS codes over finite rings leads to the fact that GRS codes over commutative rings have better parameters than their noncommutative counterparts. Also, GRS codes over finite fields have better parameters than their commutative rings counterparts. But we also show that given a unique decoding algorithm for a GRS code over a finite field, there exists a unique decoding algorithm for a GRS code over a truncated power series ring with a better asymptotic complexity. Moreover, we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is, for example, a Galois ring or a truncated power series ring or the ring of square matrices over the latter ring.

In this paper we extend several classical results on pointed torsion theories – also known as torsion pairs – to the setting of non-pointed torsion theories defined via kernels and cokernels relative to a fixed class of trivial objects (often referred to as pretorsion theories ). Our results are developed in the recently introduced framework of (non-pointed) prenormal categories and other related contexts. Within these settings, we recover some characterisations of torsion and torsion-free subcategories, as well as the classical correspondences between torsion theories and closure operators. We also suitably extend a correspondence between torsion theories and (stable) factorisation systems on the ambient category, known in the homological case. Some of these results are then further specialised to an appropriate notion of hereditary torsion theory. Finally, we apply the developed theory to construct new examples of pretorsion theories.

In this paper, we introduce the concept of S‐‐ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of ‐ideals apply to S‐‐ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m‐system, we define right S‐‐ideals. We demonstrate the equivalence of S‐‐ideals and right S‐‐ideals in commutative rings with identity and provide examples to illustrate the connections between right S‐prime ideals and ‐ideals.

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative rings this sum is nilpotent. In between commutative and non-commutative rings there are several types of rings in which this property holds. For instance, reduced, NI, AI (or IFP), 2-primal, reversible and symmetric, etc. We may term these types of rings as nearby commutative rings (in short NC-rings). In this work we have studied properties and various characterizations of such rings as well as rngs. As applications, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right Ck-Goldie rings.

This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory (T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free cover of the direct sum of the modules. The concept of aT-near homomorphism, which generalizes Enochs’ definition of a neat submodule, is introduced and studied. This allows the generalization of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free cover are called universally covering. If the inclusion map ofR into the appropriate quotient ringQ is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can be reduced to the caseR=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product ofF-neat homomorphisms is alwaysT-neat; (2) the product of torsion-free covers is alwaysT-neat; (3) every nonzero module inT has a nonzero socle.

The purpose of this paper is to give a new proof of the Joyal-Tierney theorem (unpublished), which asserts that a morphism f : R → S of commutative rings is an effective descent morphism for modules if and only if f is pure as a morphism of R-modules. Let R be a commutative ring with unit and R−mod the category of R-modules. Since, for any R-module M , the group C(M) = HomAb(M,Q/Z) (where Ab is the category of abelian groups and Q/Z is the rational circle abelian group) becomes an R-module with the action of R on C(M) by (rf)(m) = f(rm) , we can define a functor C : (R−mod)op → R−mod, given by C(M) = HomAb(M,Q/Z). Since the abelian group Q/Z is an injective cogenerator in the category of abelian groups (see, for example, [1]), the functor C is exact and reflects isomorphisms. We say that a morphism f : M → M ′ of R-modules is pure if for any R-module N , 1N ⊗R f : N ⊗R M → N ⊗R M ′ is monic. Let f : M → M ′ be a morphism of R-modules. The next lemma follows from the commutativity of the diagram HomR(C(M), C(M ′)) ≈ HomR(C(M),C(f)) HomR(C(M,C(M))) ≈ C(C(M)⊗R M ′) C(1C(M)⊗Rf) C(C(M)⊗R M), where the vertical morphisms are the canonical isomorphisms. 1. Lemma. Let f : M → M ′ be a morphism of R-modules. The following conditions are equivalent: (a) f is a pure morphism of R-modules. (b) C(f) is a split epimorphism of R-modules. Let f : R → S be a morphism of commutative rings. Recall that a descent datum on an object M ∈ Ob(S−mod) can be described as an S-module morphism θ : M → S⊗RM such that θ makes Received by the editors 1999 August 9 and, in revised form, 2000 February 2. Published on 2000 March 9. 2000 Mathematics Subject Classification: 13C99,18A20,18A30,18A40.

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T π-compact provided G/T πG is π-complete, extending the abelian group result in a natural way.

Throughout this paper it is assumed that rings are associative, have the identity element, and all modules are left unital. R will denote a ring with identity, R-Mod the category of left R-modules, and for each left R-module M, E(M) (resp. J(M)) will represent the injective hull (resp. Jacobson radical) of M. Also, for a module M, A ⊆' M will mean that A is an essential submodule of M, and Z(M) denotes the singular submodule of M. M is called singular if Z(M) = M, and it is called non-singular in case Z(M) = 0. For fundamental definitions and results related to torsion theories, we refer to [12] and [14]. In this paper we shall deal mainly with Goldie torsion theory. Recall that a pair (G, F) of classes of left R-modules is known as Goldie torsion theory if G is the smallest torsion class containing all modules B/A, where A ⊆' B, and the torsion free class F is precisely the class of non-singular modules.

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Let R be an associative ring with unit element. Mod-R and R-Mod will denote the categories of unitary right and left R-modules, respectively, and all modules are assumed to be in Mod-R unless otherwise specified. For all M, N ϵ Mod-R, HomR(M, N) will usually be abbreviated as [M, N]. For the definitions of basic terms, and an exposition on torsion theories in Mod-R, the reader is referred to Lambek [6]. Jans [5] has called a class of modules which is closed under submodules, direct products, homomorphic images, group extensions, and isomorphic images a TTF (torsion-torsionfree) class.

We are concerned with an algebra S over a commutative ring. Precisely S is a non-commutative ring with identity which is also a finitely generated unital R module such that r(xy) = (rx)y = x(ry) for r in R and x, y ∈ S. In section one, we assume A is a commutative, Artinian ring. Following Goro Azumaya (see (1, p. 273)), we define the canonical module F of A to be the injective hull of A modulo the Jacobson radical of A i.e. F = I(A/J(A)).

Constructive arithmetics in Ore localizations enjoying enough commutativity