Cotorsion Theories and Colocalization

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules $$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$ such that F0,..., Fn−1 are finitely generated free and Kn−1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext n R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor R n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

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.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.

Associative rings with metabelian adjoint group

Suppose R is a commutative Noeheian ring with maximal ideal P and E is the injective hull of R/P. The hereditary torsion theory associated with P has E has as an injective cogenerator and the associated localization is exact. Matlis [3] has shown that E Artinian. In this case it can be shown that the functor associated wity localization and then completions is exact on finitely generated modules. In this paper we examine completions of modules over noncommutative rings. We show that if an injective congenerator for a torsion theory is Artinian and the associated localization is exact then the functor given by localization and then completion is also exact. Throughout this paper we will assume that all rings are associative with identity. If R is a ring we denote RM as the category of all unital left R-modules. We will adopt the convention of writing all module homomorphisms opposite their scalars. We refer the reader to [5] and [1] for the basic results about torsion theories and kernel functors.

In this paper we introduce and investigate a class of those rings in which every finitely presented proper ideal is regular. We establish the transfer of this notion to the trivial ring extension, direct product and homomorphic image, and then generate new and original families of rings satisfying this property.

Varieties of algebras

The notion of weakly [Formula: see text]-absorbing ideals, introduced as a natural generalization of prime ideals, has gained significant attention in commutative rings. A weakly [Formula: see text]-absorbing ideal of a commutative ring [Formula: see text] ensures that for all [Formula: see text], the inclusion [Formula: see text] and [Formula: see text] implies that at least one of [Formula: see text] or [Formula: see text] belongs to [Formula: see text]. This paper builds upon the existing framework by introducing and exploring weakly [Formula: see text]-[Formula: see text]-absorbing ideals in the context of a multiplicatively closed subset [Formula: see text] of [Formula: see text]. An ideal [Formula: see text], disjoint from [Formula: see text], is defined as weakly [Formula: see text]-[Formula: see text]-absorbing if [Formula: see text] and [Formula: see text] guarantee that [Formula: see text], [Formula: see text], or [Formula: see text] for some [Formula: see text]. We give the fundamental properties of weakly [Formula: see text]-[Formula: see text]-absorbing ideals and investigate their behavior under various ring constructions, such as amalgamation duplications, homomorphic images, direct products and trivial extensions. Notable results include conditions under which the ideals retain the weakly [Formula: see text]-[Formula: see text]-absorbing property and characterizations of these ideals in special commutative rings such as reduced rings and local semiprime rings which are not prime.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

Torsion theories over commutative rings

In this article it is proven that two systems are I/O equivalent with respect to an initial state pair if and only if their minimizations are isomorphic images, which means that the minimizations are essentially the same systems with just a renaming of states, inputs, and outputs. The concept of homomorphic images is also defined: One system is a homomorphic image of the other if the one system is simpler than the other, but has essentially the same overall functionality. Two systems that are I/O equivalent with respect to an initial state pair are not necessarily homomorphic images. Hence, a system that implements the one may not implement the other. To assume otherwise can lead to disaster. It is the responsibility of systems engineers assigned to system functional analysis to consider the I/O requirement and the performance requirement (which may include reusability considerations) and to specify functional system designs for implementation. It is the responsibility of systems engineers assigned to system architecture to consider the technology requirement and the cost requirement and to specify buildable system designs to implement those exact functional system designs, not the requirements for I/O behavior. This paper shows that the equivalence of two systems cannot be proven by looking only at input and output behavior. © 2000 John Wiley & Sons, Inc. Syst Eng 3: 82–95, 2000

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

In any variety of bounded integral residuated lattice-ordered commutative monoids (bounded residuated lattices for short) the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with the property "to have radical term", especially for k-radical varieties, and for the hierarchy of varieties (WL k)k>0 defined in Cignoli and Torrens (Studia Logica 100:1107---1136, 2012 [7]).