Jae Keol Park

PRINCIPALLY QUASI-BAER RINGS

We say a ring with unity is right principally quasi-Baer (or simply, right p.q.-Baer) if the right annihilator of a principal right ideal is generated (as a right ideal) by an idempotent. This class of rings includes the biregular rings and is closed under direct products and Morita invariance. The 2-by-2 formal upper triangular matrix rings of this class are characterized. Connections to related classes of rings (e.g., right PP, Baer, quasi-Baer, right FPF, right GFC, etc.) are investigated. Examples to illustrate and delimit the theory are provided.

Polynomial extensions of Baer and quasi-Baer rings

Abstract A ring R is called (quasi-) Baer if the right annihilator of every (ideal) nonempty subset of R is generated, as a right ideal, by an idempotent of R . Armendariz has shown that for a reduced ring R (i.e., R has no nonzero nilpotent elements), R is Baer if and only if R [ x ] is Baer. In this paper, we show that for many polynomial extensions (including formal power series, Laurent polynomials, and Laurent series), a ring R is quasi-Baer if and only if the polynomial extension over R is quasi-Baer. As a consequence, we obtain a generalization of Armendarizs result for several types of polynomial extensions over reduced rings.

Triangular matrix representations of ring extensions

In this paper we investigate the class of piecewise prime, PWP, rings which properly includes all piecewise domains (hence all right hereditary rings which are semiprimary or right Noetherian). For a PWP ring we determine a large class of ring extensions which have a generalized triangular matrix representation for which the diagonal rings are prime.

Regularity conditions and the simplicity of prime factor rings

R denotes a ring with unity and Nr(R) its nil radical. R is said to satisfy conditions: 1. (1) pm(Nr) if every prime ideal containing Nr(R) is maximal; 2. (2) WCI if whenever a,e ϵ R such that e = e2, eR + Nr(R) = RaR + Nr(R), and xe − ex ϵ Nr(R) for any x ϵ R, then there exists a positive integer m such that am(1 − e) ϵ amNr(R). For example, if R is right weakly π-regular or every idempotent of R is central, then R satisfies WCI. Many authors have considered the equivalence of condition pm (i.e., every prime ideal is maximal) with various generalizations of von Neumann regularity over certain classes of rings including commutative, PI, right duo, and reduced. In the context of weakly π-regular rings, we prove the following two theorems which unify and extend nontrivially many of the previously known results. Theorem I. Let R be a ring with Nr(R) completely semiprime. Then the following conditions are equivalent: (1) R is right weakly π-regular; (2) RNr(R) is right weakly π-regular and R satisfies WCI; (3) RNr(R) is biregular and R satisfies WCI; (4) for each χ ϵ R there exists a positive integer m such that R = RχmR + r(χm). Theorem II. Let R be a ring such that Nr(R) is completely semiprime and R satisfies WCI. Then the following conditions are equivalent: (1) R is right weakly π-regular; (2) RNr(R) is right weakly π-regular; (3) RNr(R) is biregular; (4) R satisfies pm(Nr); (5) if P is a prime ideal such that Nr(RP) = 0, then RP is a simple domain; (6) for each prime ideal of R such that Nr(R) ⊆ P, then P = OP.

A connection between weak regularity and the simplicity of prime factor rings

In this paper, we show that a reduced ring R is weakly regular (i.e., 12 = I for each one-sided ideal I of R ) if and only if every prime ideal is maximal. This result extends several well-known results. Moreover, we provide examples which indicate that further generalization of this result is limited. Throughout this paper R denotes an associative ring with identity. All prime ideals are assumed to be proper. The prime radical of R and the set of nilpotent elements of R are denoted by P(R) and N(R), respectively. The connection between various generalizations of von Neumann regularity and the condition that every prime ideal is maximal will be investigated. This connection has been investigated by many authors [2, 3, 5, 7, 12, 14]. The earliest result of this type seems to be by Cohen [3, Theorem 1]. Storrer [12] was able to provide the following result: If R is a commutative ring then the following are equivalent: (1) R is 7r-regular; (2) R/P(R) is regular; and (3) all prime ideals of R are maximal ideals. Fisher and Snider extended this result to P.I. rings [5, Theorem 2.3]. On the other hand, Chandran generalized Storrers result to duo rings [2, Theorem 3]. Next Hirano generalized Chandrans result to right duo rings [7, Corollary 1]. More recently the result was generalized to bounded weakly right duo rings by Yao [14, Theorem 3]. As a corollary of our main result, we show that if R/P(R) is reduced (i.e., N(R) = P(R)) then the following are equivalent: (1) R/P(R) is weakly regular; (2) R/P(R) is right weakly 7r-regular; and (3) every prime ideal of R is maximal. This result generalizes Hiranos result for right duo rings. A further consequence of our main result is that if R is reduced then R is weakly regular if and only if every prime factor ring of R is a simple domain. This result can be compared to the well-known fact that when R is reduced, then R is von Neumann regular if and only if every prime factor ring of R is a division ring. We conclude our paper with some examples which illustrate and delimit our results. Received by the editors December 8, 1992. 1991 Mathematics Subject Classification. Primary 16D30, 1 6E50; Secondary 16N60.

MODULES WITH FULLY INVARIANT SUBMODULES ESSENTIAL IN FULLY INVARIANT SUMMANDS

ABSTRACT A module M is called (strongly) FI-extending if every fully invariant submodule is essential in a (fully invariant) direct summand. The class of strongly FI-extending modules is properly contained in the class of FI-extending modules and includes all nonsingular FI-extending (hence nonsingular extending) modules and all semiprime FI-exten ding rings. In this paper we examine the behavior of the class of strongly FI-extending modules with respect to the preservation of this property in submodules, direct summands, direct sums, and endomorphism rings.

Extensions of Rings and Modules

Preliminaries and Basic Results.- Injectivity and Some of Its Generalizations.- Baer, Rickart, and Quasi-Baer Rings.- Baer, Quasi-Baer Modules, and Their Applications.- Triangular Matrix Representations and Triangular Matrix Extensions.- Matrix, Polynomial, and Group Ring Extensions.- Essential Overring Extensions - Beyond the Maximal Ring of Quotients.- Ring and Module Hulls.- Hulls of Ring Extensions.- Applications to Rings of Quotients and C* Algebras.- Open Problems and Questions.- References.- Index.

A sheaf representation of quasi-Baer rings

We give a complete characterization of a certain class of quasi-Baer rings which have a sheaf representation (by a “sheaf representation” of a ring the authors mean a sheaf representation whose base space is Spec(R) and whose stalks are the quotients R/O(P), where P is a prime ideal of R and O(P)={a∈R|aRs=0 for some s∈R⧹P}). Indeed, it is shown that a quasi-Baer ring R with a complete set of triangulating idempotents has such a sheaf representation if and only if R is a finite direct sum of prime rings. As an immediate corollary, a piecewise domain R has such a sheaf representation if and only if R is a finite direct sum of prime piecewise domains. Also it is shown that if R is a quasi-Baer ring, then R/O(P) is a right ring of fractions; in addition, if R is neither prime nor essentially nilpotent then R has a nontrivial representation as a subdirect product of the rings R/O(P), where P varies through the minimal prime ideals of R.

Semicentral Reduced Algebras

An idempotent e of an algebra R is left semicentral if Re = eRe. If 0 and 1 are the only left semicentral idempotents in R, then R is called semicentral reduced. Recent results on generalized triangular matrix algebras and semicentral reduced algebras are surveyed. New results are provided for endomorphism algebras of modules and for semicentral reduced algebras. In particular, semicentral reduced rings which are right FPF, right nonsingular, or left perfect are described.

Principally Quasi-Baer Ring Hulls

We show the existence of principally (and finitely generated) right FI-extending right ring hulls for semiprime rings. From this result, we prove that right principally quasi-Baer (i.e., right p.q.-Baer) right ring hulls always exist for semiprime rings. This existence of right p.q.-Baer right ring hull for a semiprime ring unifies the result by Burgess and Raphael on the existence of a closely related unique smallest overring for a von Neumann regular ring with bounded index and the result of Dobbs and Picavet showing the existence of a weak Baer envelope for a commutative semiprime ring. As applications, we illustrate the transference of certain properties between a semiprime ring and its right p.q.-Baer right ring hull, and we explicitly describe a structure theorem for the right p.q.-Baer right ring hull of a semiprime ring with only finitely many minimal prime ideals. The existence of PP right ring hulls for reduced rings is also obtained. Further application to ring extensions such as monoid rings, matrix, and triangular matrix rings are investigated. Moreover, examples and counterexamples are provided.