Strongly compressible modules and semiprime right goldie rings

A module M _R is defined to be strongly compressible (or SC for short) if for every essential submodule N of M, there exists X ? E(M) such that M ? X ? N. We show that a ring R is semiprime right Goldie iff R _Ris SC module iff every right ideal of R is SC module iff every submodule of each progenerator of Mod-R is SC module. As corollaries of this result, we obtain some new module-theoretic characterizations of semiprime Goldie rings, prime (right) Goldie rings and Prüfer rings, etc., etc.,respectively. And the characterization theorem of semiprime Goldie rings of López-Permouth, Rizvi and Yousif by using weakly-injective modules can be regarded as a corollary of our results.     

Generalized APP-Rings

We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ? _R (Ra) ⁿ is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation.     

On fusible rings

We answer in negative two of questions posed in Ghashghaei and McGovern. We also establish a new characterization of semiprime left Goldie rings by showing that a semiprime ring R is left Goldie iff it is regular left fusible and has finite left Goldie dimension.     

PI-rings and semiprime rings having faithful modules with krull dimension

It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent. If, moreover, the R-module M and the left ideal_R(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings. Moreover, first, a right Artinian ring R such that rad(R)²=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension. Supported by RFFR grant No. 26-93-011-1544. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997.     

满足右零化子特殊升链条件的环

本文讨论了满足右零化子特殊升链条件的环的性质以及与左、右完备环，ＱＦ－环的关系；给出了满足右零化子特殊升链条件的环中素理想是完全素理想的条件，从而给出了Ｇｏｌｄｉｅ定理的一个应用．     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

Weakly automorphism invariant modules and essential tightness

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.     

Cotorsion pairs generated by modules of bounded projective dimension

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an ℵ₀-noetherian ring Q of little finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and to Cohen Macaulay noetherian commutative rings.     

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.     

Semiprime ideals in rings with finite group actions

LetR be a ring, G a finite group of automorphisms acting on R, and R^G the-fixed subring of R. We prove that if R is semiprime with no additive ¦ G¦-torsion, then R is left Goldie if and only if R^G is left Goldie. By coupling this with an examination of the prime ideal structures of R^G and R, we are able to prove that if ¦G ¦ is invertible in R and R^G is left Noetherian, then R satisfies the-ascending chain condition on semiprime ideals, every semiprime factor ring of R is left Goldie, and nil subrings of R are nilpotent. For the pair R^G and R, we also consider various other properties of prime and maximal ideals such as lying over, going up, going down, and incomparability.     

Self-injective rings and linear (weak) inverses of linear finite automata over rings

LetR be a finite commutative ring with identity and τ be a nonnegative integer. In studying linear finite automata, one of the basic problems is how to characterize the class of rings which have the property that every (weakly) invertible linear finite automaton ℳ with delay τ over R has a linear finite automaton ℳ′ over R which is a (weak) inverse with delay τ of ℳ. The rings and linear finite automata are studied by means of modules and it is proved that *-rings are equivalent to self-injective rings, and the unsolved problem (for τ=0) is solved. Moreover, a further problem of how to characterize the class of rings which have the property that every invertible with delay τ linear finite automaton ℳ overR has a linear finite automaton ℳ′ over R which is an inverse with delay τ′ for some τ′⩾τ is studied and solved. Project supported by the National Natural Science Foundation of China(Grant No. 69773015).     

On the characterization of radical and semisimple classes in categories

We define and characterize radical and semisimple classes in a category K which satisfies certain conditions. These conditions are such that K could be any of the categories of associative rings, groupsR-modules, topological spaces or graphs. Among others, the following is proved:.

A class of objects R in K is a radical class if and only if K is a cohereditary component class which is closed under extensions and with T ? R. A class of objects S in K is a semisimple class if and only if S is a hereditary class which is closed under subdirect embed-dings and extensions with T ? S.     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

CLASSICAL QUOTIENT RINGS OF GROUP RINGS

Abstract

Throughout G will denote a free Abelian group and Z(R) the right singular ideal of a ring R. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R/Z(R) is semiprime, Z(R) is rationally closed, and Z(R) contains no closed uniform right ideals. We prove that R is a Cl-ring if and only if the group ring RG is a C1-ring. If RG has the additional property that bRG is dense whenever b is a right nonzero-divisor, then the complete ring of quotients of RG is a classical ring of quotients.     

Rings radical over subrings

LetR be a ring with a subringA such that a power of every element ofR lies inA. The following results are proved: IfR has no nonzero nil right ideals, neither doesA; if moreoverR is prime,A is also prime. IfR is semiprime Goldie, so isA. IfA has no nonzero nilpotent elements, then the nilpotent elements ofR form an ideal. Finally ifR has no nil right ideals andA is Goldie, thenR is Goldie. This work was supported by the National Research Council of Brazil (CNPq) at the University of Chicago. The author wishes to express his gratitude to Professor I. N. Herstein for his advise and encouragement.     

Semiprime rings with finite length W.R.T. an idempotent kernel functor

The purpose of this note is to show that the only idempotent kernel functor σ on Mod-R, whereR is a semiprime ring andR _R has finite σ-length, is the idempotent kernel functorZ corresponding to the Goldie torsion theory.     

Goldie Conditions for Ore Extensions over Semiprime Rings

Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ,δ] and both rings have the same left uniform dimension. Presented by S. Montgomery Mathematics Subject Classification (2000) 16S90. Jerzy Matczuk: Supported by the Flemish–Polish bilateral agreement BIL 01/31.     

Rings with all modules residually finite

Define a ringA to be RRF (resp. LRF) if every right (resp. left) A-module is residually finite. Refer to A as an RF ring if it is simultaneously RRF and LRF. The present paper is devoted to the study of the structure of RRF (resp. LRF) rings. We show that all finite rings are RF. IfA is semiprimary, we show thatA is RRF ⇔A is finite ⇔A is LRF. We prove that being RRF (resp. LRF) is a Morita invariant property. All boolean rings are RF. There are other infinite strongly regular rings which are RF. IfA/J(A) is of bounded index andA does not contain any infinite family of orthogonal idempotents we prove:

Partial Skew Polynomial Rings and Goldie Rings

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R?x; α? is right Goldie, where R[x; α] (R?x; α?) denotes the partial skew (Laurent) polynomial ring over R. In addition, R?x; α? is semiprime while R[x; α] is not necessarily semiprime.     

Finitely 1-convex f-rings

This paper investigates f-rings that can be constructed in a finite number of steps where every step consists of taking the fibre product of two f-rings, both being either a 1-convex f-ring or a fibre product obtained in an earlier step of the construction. These are the f-rings that satisfy the algebraic property that rings of continuous functions possess when the underlying topological space is finitely an F-space (i.e. has a Stone-?ech compactification that is a finite union of compact F-spaces). These f-rings are shown to be SV f-rings with bounded inversion and finite rank and, when constructed from semisimple f-rings, their maximal ideal space under the hull-kernel topology contains a dense open set of maximal ideals containing a unique minimal prime ideal. For a large class of these rings, the sum of prime, semiprime, primary and z-ideals are shown to be prime, semiprime, primary and z-ideals respectively.