Gary F. Birkenmeier

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.

Modules in Which Every Fully Invariant Submodule is Essential in a Direct Summand

Abstract A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring.

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.

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.

Goldie Extending Modules

In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

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.

Medial near-rings

In this paper we discuss (left) near-rings satisfying the identities:abcd=acbd,abc=bac, orabc=acb, called medial, left permutable, right permutable near-rings, respectively. The structure of these near-rings is investigated in terms of the additive and Lie commutators and the set of nilpotent elementsN (R). For right permutable and d.g. medial near-rings we obtain a “Binomial Theorem,” show thatN (R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. “Natural” examples from analysis and geometry are produced via a general construction method.