Arbitrary Associative Ring Research Articles

ON A FINITISTIC COTILTING-TYPE DUALITY

ABSTRACT Let R and S be arbitrary associative rings. Given a bimodule , we denote by and the functors and , where or S. We say that is a finitistic weakly cotilting bimodule (briefly FWC) if for each module M cogenerated by W, finitely generated or homomorphic image of a finite direct sum of copies of W, . We are able to describe, on a large class of finitely generated modules, the cotilting-type duality induced by a FWC-bimodule.

On a Finitistic Cotilting-Type Duality1

ABSTRACT Let R and S be arbitrary associative rings. Given a bimodule R W s , we denote by Δ? and Γ? the functors Hom?(−, W) and (−, W), where or S. We say that is a finitistic weakly cotilting bimodule (briefly FWC) if for each module M cogenerated by W, finitely generated or homomorphic image of a finite direct sum of copies of W, . We are able to describe, on a large class of finitely generated modules, the cotilting-type duality induced by a FWC-bimodule.

The Automorphism Group of Certain Radical Matrix Rings

This paper is devoted to the study of automorphisms of matrix radical rings. The area has been under active investigation since the 1950s. Automorphisms of the algebra NTn K of all (lower) niltriangular n× n matrices over a field K were described by Dubish and Perlis [1, Theorem 5-7]. It is easy to verify that the automorphism group AutR of any radical ring R coincides with the intersection of the automorphism group of the adjoint group G R and the automorphism group of the associated Lie ring R of R. The adjoint group of NTn K is isomorphic to the unitriangular group UTn K . If K is a finite field, then the group UTn K is a Sylow subgroup of GLn K and its automorphisms were studied in [13, 14, 16, 17]. For arbitrary associative ring K with identity automorphism groups of

ON THE EXISTENCE OF FLAT COVERS IN R-gr

Recently, a proof of the existence of a flat cover of any module over an arbitrary associative ring with unit has been finally given (see 4-5). In this paper we prove the existence of flat covers in the category of graded modules over a graded ring. Some graded theoretical machinery is introduced to make the proof possible and new graded homological tools are developed.

ON PRIMARY AND REDUCED VARIETIES OF MONOASSOCIATIVE ALGEBRAS

The notions of solvability, completeness (divisibility), primarity, reducibility, and purity play an important role in group theory. It turns out that there is another approach to these notions which leans on the theory of varieties of groups. This circumstance makes it possible to define analogs of these notions for arbitrary algebras. As far as solvability is concerned, this notion was introduced in [1] (also see [2]), wherein many problems were posed. Developing these ideas in [3], the author introduced the notions of primary, complete, and reduced algebras as well as that of pure subalgebra and some other notions. Some related results for modules and linear algebras were also mentioned in [3]. In the present article we characterize primary varieties of arbitrary monoassociative algebras over an arbitrary associative and commutative unital ring and characterize reduced varieties with certain properties for monoassociative algebras over Dedekind rings. In particular, these characterizations cover the cases of associative, alternative, Lie-type, and Jordan algebras (the corresponding results for rings were announced in [4]).

Generalizing Morita Duality: A Homological Approach

Let R and S be arbitrary associative rings. Given a bimodule RWS, we denote by Δ? and Γ? the functors Hom?(−,W) and Ext1?(−,W), where ?=R or S. The functors ΔR and ΔS are right adjoint with the evaluation maps δ as unities. A module M is Δ-reflexive if δM is an isomorphism. In this paper we give, for a weakly cotilting bimodule RWS, the notion of Γ-reflexivity. We construct large Abelian subcategories MR and MS where the functors ΓR and ΓS are left adjoint and a “cotilting theorem” holds.

The Boundary Equivalence for Rings and Matrix Rings over Them

We study into the question of whether some rings and their associated matrix rings have equal decidability boundaries in the scheme and scheme-alternative hierarchies. Let $$B_H (A;\sigma )$$ be a decidability boundary for an algebraic system 〈 A; σ 〉 w.r.t. the hierarchy H. For a ring R, denote by $$\underline M _n (R)$$ an algebra with universe $$\bigcup\limits_{1 \leqslant k,l \leqslant n} {R^{k \times l} } $$ . On this algebra, define the operations + and ⋅ in such a way as to extend, if necessary, the initial matrices by suitably many zero rows and columns added to the underside and to the right of each matrix, followed by “ordinary” addition and multiplication of the matrices obtained. The main results are collected in Theorems 1-3. Theorem 1 holds that if R is a division or an integral ring, and R has zero or odd characteristic, then the equalities $$\mathcal{B}_S \left( {R; + , \cdot } \right) = \mathcal{B}_S \left( {R^{n \times n} ; + , \cdot } \right){\text{ and }}\mathcal{B}_S \left( {R; + , \cdot ,1} \right) = \mathcal{B}_S \left( {R^{n \times n} ; + , \cdot ,1} \right)$$ hold for any n⩾1. And if R is an arbitrary associative ring with identity then $$\mathcal{B}_S \left( {R; + , \cdot ,1} \right) = \mathcal{B}_S \left( {R^{n \times n} ;\sigma _0 \cup \left\{ {e_{ij} } \right\}} \right)$$ for any n ⩾ 1 and i,j ∈ { 1,..., n}, where e ij is a matrix identity. Theorem 2 maintains that if R is an associative ring with identity then $$\mathcal{B}_S \left( {\underline M _n \left( R \right)} \right) = \mathcal{B}_S \left( {R; + , \cdot } \right)$$ . Theorem 3 proves that $$\mathcal{B}_{SA} \left( {\underline M _n \left( \mathbb{Z} \right)} \right) = \left\{ {\forall \neg \vee ,\exists \neg \wedge ,\forall \exists ,\exists \forall } \right\}$$ for any n ⩾ 1.

Associative rings satisfying the Engel condition

Let C C be a commutative ring, and let R R be an associative C C -algebra generated by elements { x 1 , … , x d } \{x_1,\ldots ,x_d\} . We show that if R R satisfies the Engel condition of degree n n , then R R is upper Lie nilpotent of class bounded by a function that depends only on d d and n n . We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.

The normalizer of the automorphism group of a module arising under extension of the base ring

Let Λ te an arbitrary associative ring with unity and let R be its unital subring contained in the center of Λ. Further, let M=λ M be a left free Λ-module of finite rank. In this paper, the normalizer of the subgroupAut(λM) of automorphisms of the module λM in the groupAut(RM) of automorphisms of the moduleRM is computed. If the ring Λ is additively generated by its invertible elements, then the above normalizer coincides with the semidirect product of the normal subgroupAut(λM) and a subgroup isomorphic to the groupAut(Λ/R) of all ring automorphisms of the ring Λ that are identical on R. Bibliography: 1 title.

Every ∗-Module Is Finitely Generated

We solve one of the main problems of the theory of representable equivalences of module categories: we prove that every ∗-module over an arbitrary associative ring R with unit is finitely generated. Consequently, we obtain the equality of the classes of all ∗-modules and all quasi-progenerators provided that R is commutative.

The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)

The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.

Normal net subgroups of the full linear group

We prove that for n ≥ 3 a net subgroup of the full linear group G=GL(n, Λ) over an arbitrary associative ring Λ with unity (see [1]) is normal in G if and only if it is a principal congruence subgroup. We also study the case n=2, where the situation is, in general, more complicated.

Primary decompositions and morita contexts

As shown by Mueller, a Morita context (R,M,N,S) yields a one-to-one order preserving correspondence between certain torsion theories on RMand SMIn addition, Goldman has defined the notions of prime kernei functors and primary modules for arbitrary associative rings. These notions lead to a type of primary decomposition for kernel functors. In this paper we show that under the above correspondence prime kernel functors correspond to prime kernel functors and primary modules correspond to primary modules. Finally we show that the above correspondence preserves the primary decomposition of kernel functors.

The Torsion Submodule of A Cyclic Module Splits Off

A prominent question in the study of modules over an integral domain has been: “When is the torsion submodule t(A) of a module A a direct summand of A?” A module is said to split when its torsion module is a direct summand. Clearly, every cyclic module over an integral domain splits. Interesting splitting problems have been explored by Kaplansky [14; 15], Rotman [20], Chase [4], and others.Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Two of the most important of these concepts are Goldie's torsion theory (see [1; 12; 22]) and the simple torsion theory (see [5; 6; 8; 9; 23], and their references).

A Construction in General Radical Theory

Given an arbitrary associative ring R we consider the ring R[x] of polynomials over R in the commutative indeterminate x. For each radical property S we define the function S* which assigns to each ring R the idealof R. It is shown that the property SA (that a ring R be equal to S*(R)) is a radical property. If S is semiprime, then SA is semiprime also. If S is a special radical, then SA is a special radical. SA is always contained in S. A necessary and sufficient condition that S and SA coincide is given.The results are generalized in the last section to include extensions of R other than R[x], One such extension is the semigroup ring R[A], where A is a semigroup with an identity adjoined.

The torsion submodule splits off

A classical question for modules over an integral domain is, "When is the torsion submodule t(A) of a module A a direct summand of AT' A module is said to split when its torsion submodule is a direct summand. Kaplansky has shown [-14] that if R is a Dedekind domain, then every module whose torsion submodule is of bounded order splits. The converse of this result has been shown by Chase [5]. Results of [15] and [3] show that every finitely generated module splits if and only if R is a Priifer domain. Finally, if every R-module splits, then Rotman has shown [19] that R must be a field. Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Almost all of these are special cases of "torsion theories" in the sense of Dickson [6]. Moreover, most of these torsion theories are hereditary (i.e., the submodule of a torsion module is torsion); and hereditary torsion classes are classes of negligible modules associated with a topologizing and idempotent filter of left ideals in the sense of Gabriel [12] (also see [17]). Some recent papers ([4], [7], and [11]) have dealt with splitting results for specific hereditary torsion theories over certain commutative rings. The main purpose of this paper is to continue the investigation of the splitting properties of a torsion theory of modules over a commutative ring. Some characterizations for the splitting of modules, whose torsion submodules have bounded order, are obtained (see definition of bounded order below and Theorems 2.2 and 4.6). In particular, these results generalize the abovementioned theorems of Chase [5] and Kaplansky [14]. Our results show that the splitting of modules whose torsion submodules have bounded order frequently forces non-zero torsion modules to have non-zero socles. This increases our interest in the smallest hereditary torsion class 6e containing the simple modules. The class ow has previously been used in the study of commutative Noetherian rings and (left) perfect rings (e.g., see [7] and [10] and their references). For a commutative ring R, we show that 6e(A) is a summand of each R-module A if and only if non-zero R-modules have non-zero socles. This generalizes the main results of [7] and [11]; moreover, in case R is a Dedekind domain, our result coincides with the above result of Rotman. We also examine the properties of a splitting hereditary torsion theory of modules over a local ring R (unique maximal left ideal). We show that if R possesses a non-trivial torsion theory (~--, #-) such that every finitely generated module splits, then R is an integral domain, and (Y-, ~ is Goldie's torsion

Lower Radicals in Associative Rings

Given a homomorphically closed class of (not necessarily associative) rings , the lower radical property determined by is the least radical property for which all rings in are radical. Recently (7) a process of constructing the lower radical property from a class of associative rings has been given which terminates after a countable number of steps. In this process, an ascending chain of classesis obtained and the property of being a ring in the class is the lower radical property determined by . In Theorem 1 we give another characterization of the rings in the class , λ ∈ {1, 2, …, omega;0}, and a procedure for constructing the lower radical determined by in an arbitrary associative ring is given.

III.—Multiplicative Ideal Theory and Rings of Quotients. I.

SynopsisBy means of a generalized ring of quotients multiplicative ideal theory is studied in an arbitrary (associative) ring. A suitable generalization of the concept of maximal order is given and factorization theorems are obtained for the nonsingular (two sided) ideals, which generalize the theorems of Artin and E. Noether.

Semisimple maximal quotient rings

Notation and Introduction. R denotes an arbitrary associative ring. A right R-module A over R will be denoted AR. BR is a large submodule of AR (AR is an essential extension of BR), if BR is a submodule of AR having nonzero intersection with every nonzero submodule of AR. A right ideal I of R is a large right ideal, if IR is a large submodule of RR. Given AR, Z(AR) is the singular submodule of AR [9], which consists of all those elements of A whose annihilators in R are large right ideals. Following Johnson [9], Q is a right quotient ring of R if Q is a ring with identity containing R as a subring (the identity of Q is the identity of R if R has one) and RR is a large submodule of QR. The quotient rings considered by Goldie in [6], [7] will be called classical quotient rings. Q is a classical right quotient of R if every regular element (nonzero divisor) of R is a unit in Q and every element of Q is of the form ab -1, a, b E R, b regular in R. In general, a ring R need not possess a classical right quotient ring. Goldie [7], has given necessary and sufficient conditions that a ring possess a classical right quotient ring which is semisimple. Here semisimple means semisimple with minimum condition [8]. This paper is concerned with the question of characterizing those rings which have a semisimple maximal right quotient ring [4], [9], [10], [11] and in this case generalizing some simple well-known results about commutative integral domains, their quotient rings and modules over these domains. Johnson [9] has shown that R has a regular maximal right quotient ring Q if and only if Z(RR) = 0, where Q is a regular ring [13] if every finitely generated right (left) ideal of Q is generated by an idempotent. In this case QR is injective [3] as a right R-module, hence the injective hull of R [2]. A ring R has a semisimple maximal right quotient ring Q if and only if Z(RR) =O and dim RR is finite, where a right R-module M is of finite dimension if every direct sum of submodules of M has only finitely many nonzero summands. This is the main result of ?1. In addition another characterization is given for rings which possess a semisimple classical right quotient ring, namely, R has a semisimple classical right quotient ring if and only if Z(RR) = 0 and if I is a large right ideal of R, then there is an element a E I such that aR is a large right ideal of R. If R has a semisimple classical right quotient ring Q, then it is known [3], that