Reversible Group Rings Over Commutative Rings

Let G be a torsion group and R be a commutative ring with identity. We investigate reversible group rings RG over commutative rings, extending results of Gutan and Kisielewicz which characterize all reversible group rings over fields.     

A krull-like dimension for noncommutative rings

LetR be an associative ring. Using chains of prime torsion theories over the category of unitary leftR-modules, a generalization of the Krull dimension for commutative rings is defined and its properties investigated.     

Completions of commutative rings and modules

In this work we continue studying the notion of completion ofR-modules, over a commutative ringR, relative to a torsion theoryϑ. We develop some techniques relative to localization at prime ideals and give structural results on the completion of finitely generatedR-modules, describing it as the product of classical completions on local noetherian rings. The authors acknowledge partial support from the D.G.I.C.Y T.     

The isomorphism relation between tree-automatic Structures

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ₂¹-set nor a Π₂¹-set.     

环的约化群与群环上的投射模

本文系统地研究群环的约化群，利用约化群刻划了群环上模的结构。主要结果：（１）Ｒ为交换半遗传环且Ｋ＿０Ｒ为挠群ｉｆｆ对任何有限生成半自反Ｒ－模Ｐ，ｓ＞０，使得．（２）设Ｒ为半局部Ｄｅｄｅｋｉｎｄ环，Ｇ为有限生成Ａｂｅｌ群，则Ｋ＿０ＲＧ为挠群ｉｆｆ如果Ｇ有素数ｐ阶元，则（３）如果Ｋ＿０ＲＧ为挠群，［Ｇ∶Ｈ］＜∞，则对任何，有．这里Ｒ为整环，Ｌ为其分式域。     

FINITE RINGS WITH COMMUTING NILPOTENT ELEMENTS

As a generalization of Wedderburn's theorem, Herstein [5] proved that a finite ring R is commutative, if all nilpotent elements are contained in the center of R. However a finite ring with commuting nilpotent elements is not necessarily commutative. Recently, in [9] and [10], Simons described the structure of finite rings R with J(R)² = 0 in a variety with definable principal congruences. In this paper, we will consider the difference between the finite commutative rings and the finite rings in which any two nilpotent elements commute with each other. As a consequence, we describe the structure of finite rings R with [J(R), J(R)] = 0 in a variety with definable principal congruences.     

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.     

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops.

Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M ₀(Q) such that Q/M ₀(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C ₀(Q) such that Q/C ₀(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6.

We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.     

Envelopes of commutative rings

Given a significative class ℱ of commutative rings, we study the precise conditions under which a commutative ring R has an ℱ-envelope. A full answer is obtained when ℱ is the class of fields, semisimple commutative rings or integral domains. When ℱ is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.     

Commutative Non-Noetherian Rings with the Diamond Property

A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.

     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

Homological Characterizations of Rings with Property (P)

Abstract

A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and Π-coherent rings with property (P).     

On Generalizations of Commutativity

This note is concerned with generalizations of commutativity. We introduce identity-symmetric and right near-commutative, and study basic structures of rings with such ring properties. It is shown that if R is an identity-symmetric ring, then the set of all nilpotent elements forms a commutative subring of R. Moreover, identity-symmetric regular rings are proved to be commutative. The near-commutativity is shown to be not left-right symmetric, and we study some conditions under which the near-commutativity is left-right symmetric. We also examine the near-commutativity of skew-trivial extensions, which has a role in this note.     

Baer modules over valuation domains

Summary A module B over a commutative domain R is said to be a Baer module if Ext _R ¹ (B, T)=0for all torsion R-modules T. The case in which R is an arbitrary valuation domain is investigated, and it is shown that in this case Baer modules are necessarily free. The method employed is totally different from Griffith's method for R=Z which breaks down for non-hereditary rings.This research was partially supported by NSF Grants DMS-8400451 and DMS-8500933.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Idempotents and the multiplicative group of some totally bounded rings

In this paper, we extend some results of D.Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power 2^ℵ0 commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.     

On the Witt ring and some arithmetical invariants of quadratic forms

The Witt ring of a field serves as an effective medium to study certain arithmetical invariants of quadratic forms, such as: s = the Stufe (the least number of summands to represent ?1 as a sum of squares), q = the number of square classes, u = the maximal anisotropic dimension of a quadratic form over the given field, and h = the height (the minimal 2-power that kills the torsion subgroup of the Witt group). These invariants may also be defined over commutative rings. This paper discusses these invariants and extend the investigations to some commutative rings, e.g. valuation rings, connected semilocal rings, Prüfer rings.     

Maximally Prüfer Rings

In this paper we consider six Prüfer-like conditions on a commutative ring R, and introduce seventh condition by defining the ring R to be maximally Prüfer if R _M is Prüfer for every maximal ideal M of R, and we show that the class of such rings lie properly between Prüfer rings and locally Prüfer rings. We give a characterization of such rings in terms of the total quotient ring and the core of the regular maximal ideals. We also find a relationship of such rings with strong Prüfer rings.