Gary F. Birkenmeier

• Ph.D
• Professor (Full) at University of Louisiana at Lafayette

Let N be a submodule of a right R-module MR, and H=End(MR). Then N is said to be projection invariant in M, denoted by N⊴pM, if eN⊆N for all e=e2∈H. We call MR π-endo Baer, denoted π-e.Baer, if for each N⊴pM there exists e=e2∈H such that lH(N)=He where lH(N) denotes the left annihilator of N in H. We show that this class of modules lies strictly be...

This is the first in a series of papers on the classification of indecomposable QF-rings. Our classification is based on the concepts of a nilary ring and the essentiality of the traces of idempotent generated right ideals. Recall that a ring R is called nilary if AB = 0 then A is nilpotent or B is nilpotent for all ideals A and B of R. In this pap...

All groups considered are Abelian. As is well known, in a divisible group every subgroup is essential in a direct summand. Moreover, the fully invariant subgroups play a crucial role in the structure of an Abelian group. Thus it is natural to consider the class of groups in which every fully invariant subgroup is essential in a direct summand. In t...

We define a ring R to be right cP-Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. This class of rings generalizes the class of right p.q.-Baer rings. We also define a ring R to be right I-extending if each ideal generated by an idempotent is essential, as a right R-module, in a right ideal gen...

In this paper, we survey results involving the projection invariant condition on one-sided ideals of rings. We focus on rings satisfying the right projection invariant extending condition (denoted right \(\pi \)-extending) or the projection invariant Baer condition (denoted \(\pi \)-Baer). Examples are provided to illustrate and delimit the results...

Idempotents dominate the structure theory of rings. The Peirce decomposition induced by an idempotent provides a natural environment for defining and classifying new types of rings. This point of view offers a way to unify and to expand the classical theory of semiperfect rings and idempotents to much larger classes of rings. Examples and applicati...

Idempotents dominate the structure theory of rings. The Peirce decomposition induced by an idempotent provides a natural environment for defining and classifying new types of rings. This point of view offers a way to unify and to expand the classical theory of semiperfect rings and idempotents to much larger classes of rings. Examples and applicati...

In this paper we show that for a given set of pairwise comaximal ideals $\{X_i\}_{i\in I}$ in a ring $R$ with unity and any right $R$-module $M$ with generating set $Y$ and $C(X_i)=\sum\limits_{k\in\mathbb{N}}\underline{\ell}_M(X_i^{k})$, $M=\oplus_{i\in I}C(X_i)$ if and only if for every $y\in Y$ there exists a nonempty finite subset $J\subseteq I...

We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Pierce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triagular generalized matrix rings are pertinent examples of the class of rings which we study. The triviality of the partic...

Let be a nonempty subset of the set of submodules of a module M. Then M is called a -extending module if for each X in there exists a direct summand D of M such that X is essential in D. In general, it is known that an essential extension of a -extending module is not -extending. In this paper, our goal is to show how to construct essential extensi...

This addendum and corrigendum is written to correct the proof of Theorem 3.2 and to correct the statement and proof of Proposition 3.4 of Birkenmeier and Tercan [2]. We add Proposition A to give a new characterization of the CH condition.

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for...

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete s...

A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module -extending if every member of the set is essential in a direct summand where is a subset of the set of all submodules of M. Our focus is the behavior of the -extending modules with respect to direct sums and di...

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi- Baer (hence all Baer) rings and all right...

In this paper the idea of a dense intrinsic extension of a ring is introduced and studied in detail. These types of extensions provide a natural generalization of the usual notion of a dense extension. Several important properties transfer to dense intrinsic extensions which include extending, quasi-continuous, and the Kasch property amongst others...

The first-named author is grateful for the hospitality provided by Hacettepe University. Abstract. A module is called an extending (or CS) module if every submodule is essential in a direct summand of the module. In this survey, we consider various generalizations of the extending property that the authors have developed over the last 25 years. To...

The focus of this monograph is the study of rings and modules which have a rich supply of direct summands with respect to various extensions. The first four chapters of the book discuss rings and modules which generalize injectivity (e.g., extending modules), or for which certain annihilators become direct summands (e.g., Baer rings). Ring extensio...

Recall that in a commutative ring R an ideal I is called primary if whenever a,b∈Ra,b∈R with ab∈Iab∈I then either a∈Ia∈I or bn∈Ibn∈I, for some positive integer n. A commutative ring R is called primary if the zero ideal is a primary ideal. In this paper, we investigate various generalizations of the primary concept to noncommutative rings. In parti...

Transference of the ring properties discussed in previous chapters to various ring extensions is the focus of this chapter. Results developed earlier are utilized to do so. It is observed that the Baer property of a ring does not transfer to its ring extensions so readily and that this happens only under special conditions. However, it will be show...

The focus of the final chapter of the book is on applications of the ideas and results developed in earlier chapters to Functional Analysis and Ring Theory. The chapter begins with the development of necessary and sufficient conditions on a ring so that its maximal right ring of quotients can be decomposed into a direct products of indecomposable r...

This chapter is devoted to properties of rings for which certain annihilators are direct summands. Such classes of rings include those of Baer rings, right Rickart rings, quasi-Baer rings, and right p.q.-Baer rings. The results and the material presented in this chapter will be instrumental in developing the subject of our study in later chapters....

In this chapter, we present a list of open problems and questions to stimulate further research on the material discussed in this monograph.

The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion o...

In this beginning chapter of the book, basic notions, definitions, terminology, and notations used throughout the book are presented. Preliminary results and related material have been included for the convenience of the reader.

The focus of this chapter is on right essential overrings of a ring which are not right rings of quotients. Osofsky’s well-known example of a finite ring whose injective hull has no compatible ring structure is considered and generalized. All possible right essential overrings of the ring in Osofsky’s example are discussed. A ring R is constructed...

Conditions which generalize injective modules and which relate to this notion are presented in this chapter. The notions of extending, quasi-continuous and continuous modules are discussed and a number of their properties are included. It is known that direct summands of modules satisfying either of (C1), (C2) or (C3) conditions, of (quasi-)injecti...

The existence and usefulness of the injective hull of a module is well known. In this chapter several hulls for a ring or a module which are essential extensions that are “minimal”, in some sense, with respect to being contained in some designated class of rings or modules are introduced. The definition of hulls includes most of the known hulls (e....

We extend various properties from a direct summand X of a module M, whose complement is semisimple, to its trace in M or to M itself. The case when MR = RR and the properties are injectivity or P-injectivity is fully described. As applications, we extend some known results for right HI-rings and give a new characterization of semisimple rings. We c...

In this corrigendum to the paper, “Goldie*-supplemented modules” [1], we present revised Propositions 3.7 and 3.12 and correct some typographical errors.

In this paper, we characterize [Formula: see text]-extending (Goldie extending) modules over Dedekind domains and we use the [Formula: see text]-extending condition to characterize the modules over a principal ideal domain whose pure submodules are direct summands. Moreover, we show that if R is a principal ideal domain, then the class of [Formula:...

The quasi-Baer condition of R/P(R) is investigated when R is a quasi-Baer ring, where P(R) is the prime radical of R. We provide an example of quasi-Baer ring R such that R/P(R) is not quasi-Baer. However, when P(R) is nilpotent, we prove that if R is a quasi-Baer (resp., Baer) ring, then R/P(R) is quasi-Baer (resp., Baer). Examples which illustrat...

A module M is said to be extending (𝒢-extending) if for each submodule X of M there exists a direct summand D of M such that X is essential in D (X ∩ D is essential in both X and D). It is known that for a nonsingular module the concepts of 𝒢-extending and extending coincide. However, in the not nonsingular case, they are distinct. In this article,...

In this article, we investigate the 𝒢-extending condition under various ring extensions. We show that if RR is 𝒢-extending and S is a right essential overring, then SR and SS are 𝒢-extending. For split-null extensions, we show that if M ⊴ R and M is left faithful, then RR is (𝒢-) extending if and only if SS is (𝒢-) extending, where S = S(R, M). Thi...

In this paper we characterize internally a TSA ring (i.e. a generalized triangular matrix ring with simple Artinian rings on the diagonal) in terms of its prime ideals. Also we show that the class of semiprimary quasi-Baer rings is a proper subclass of the class of TSA rings. Moreover, we generalize results of Harada, Small, and Teply on semiprimar...

Let R be a ring and nil(R) the set of all nilpotent elements of R. For a subset X of a ring R, we define N R(X) = {a ∈R {pipe}xa ∈ nil(R) for all x ∈ X}, which is called the weak annihilator of X in R. In this paper we mainly investigate the properties of the weak annihilator over extension rings.

In this expository paper, we survey results on the concept of a hull of a ring or a module with respect to a specific class of rings or modules. A hull is a ring or a module which is minimal among essential overrings or essential overmodules from a specific class of rings or modules, respectively. We begin with a brief history highlighting various...

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 Bu...

This corrigendum is written to correct the proof of Theorem 5.3 of Akalan et al. [11. Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 . [Taylor & Francis Online], [Web of Science ®]View all references].

We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-...

Motivated by a relation on submodules of a module used by both A. W. Goldie and P. F. Smith, we say submodules X, Y of M are β* equivalent, Xβ*Y, if and only if is small in and is small in . We show that the β* relation is an equivalence relation and has good behaviour with respect to addition of submodules, homomorphisms and supplements. We apply...

This volume features a collection of 13 peer-reviewed contributions consisting of expository/survey articles and research papers by 24 authors. Many of theses contributions were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, from August 18 to 22, 2008. The selected contributions e...

In this paper, we extend various classical results by Armendariz and Steinberg, Fisher, Kaplansky, Martindale, Posner, and Rowen on semiprime PI-rings. We do this by introducing several new generalizations of the class of semiprime PI-rings. For these new classes, some structure theorems are obtained, and connections to arbitrary semiprime rings ar...

For a ring R, we investigate and determine “minimal” right essential overrings (right ring hulls) belonging to certain classes which are generated by R and subsets of the central idempotents of Q(R), where Q(R) is the maximal right ring of quotients of R. We show the existence of and characterize a quasi-Baer hull and a right FI-extending hull for...

It is well known from Osofsky’s work that the injective hull E(RR) of a ring R need not have a ring structure compatible with its R-module scalar multiplication. A closely related question is: if E(RR) has a ring structure and its multiplication extends its R-module scalar multiplication, must the ring structure be unique? In this paper, we utilize...

For an arbitrary ring R we completely characterize when Q(R), the maximal right ring of quotients of R, is a direct product of indecomposable rings and when Q(R) is a direct product of prime rings in terms of conditions on ideals of R. Our work generalizes decomposition results of Goodearl for a von Neumann regular right self-injective ring and of...

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that , where is the smallest right FI-extending right ring of quotients...

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 extendi...

We investigate connections between the right FI-extending right ring hulls of semiprime homomorphic images of a ring R and the right FI-extending right rings of quotients of R by considering ideals of R which are essentially closed and contain the prime radical P(R). As an application of our results, we show that the bounded central closure of a un...

For any right essential overring T of a right FI-extending ring R, it is shown that dim(T) ≤ dim(R), where dim(−) is triangulating dimension of a ring. As a consequence, we show that for a ring R the maximal right ring of quotients, Q(R), is a direct product of finitely many prime rings if and only if Q(R) is semiprime and dim(Q(R)) is finite. Some...

We construct a ring R with R = Q(R), the maximal right ring of quotients of R, and a right R-module essential extension S R of R R such that S has several distinct isomorphism classes of compatible ring structures. It is shown that under one class of these compatible ring structures, the ring S is not a QF-ring (in fact S is not even a right FI-ext...

In this paper we investigate the radical properties of classes of rings constructed using a class pair (M1 : M2) of rings. Our theory encompasses the theory of radical pairs and allows the study of ring theoretical questions in terms of radicals. For instance, Hilbert's Nullstellensatz and an equivalent form of Koethe's conjecture can be stated in...

A module M is said to satisfy the C 11 condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C 1 condition implies the C 11 condition and that the class of C 11-modules is closed under direct sums but not under direct summands. We show that if M = M 1 M 2, where M has C 11 and M 1 is...

Without Abstract

Our goal is to develop methods that enable one to select a class K of rings and then to describe all right essential overrings or all right rings of quotients of a given ring R which lie in K. Our major method of attack is to determine the existence and/or uniqueness of right ring hulls of R in K and to use these to characterize the right essential...

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 ring...

Snider showed that the class of radicals of rings has a natural lattice structure. The same is true for any universal class of near-rings. We show that the classes of hereditary and ideal-hereditary radicals, inter alia, are complete sublattices. Atoms of certain sublattices are discussed. If N is a universal class of near-rings containing the univ...

An abelian group has the FI-extending property if every fully invariant subgroup is essential in a direct summand. A mixed abelian group has the FI-extending property if and only if it is a direct sum of a torsion and a torsion-free abelian group, both with the FI-extending property. A full characterization is obtained for the abelian groups with t...

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 thi...

A right R-module M has right SIP (SSP) if the intersection (sum) of two direct summands of M is also a direct summand. It is shown that the right SIP (SSP) is not a Morita invariant property and that a nonsingular C 11+-module does not necessarily have SIP. In contrast, it is shown that the direct sum of two copies of a right Ore domain has SIP as...

We extend various properties from a direct summand X of a module M, whose complement is semisimple, to its trace in M or to M itself. The case when M R = RR and the properties are injectivity or P-injectivity is fully described. As applications, we extend some known results for right HI-rings and give a new characterization of semisimple rings. We...

In this survey, we provide some results and examples on the behavior of the quasi-Baer and the right FI-extending right ring hulls. We focus on these ring hulls for various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. We also establish connections between the right...

In this paper, we continue the study of various annihilator conditions which were used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. In our main results, we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of polynomia...

A class K of rings has the GADS property (i.e., generalized ADS property) if wheneverX◃ I◃ R with X∈ K, then there exists B ◃ R with B ∈ K such that X ⊆ B ⊆ I. Radicals whose semisimple classes have the GADS property are called g-radicals. In this paper, we fully characterize the class of g -radicals. We show that ? is a g-radical if...

In this chapter, generalized triangular matrix representations are discussed by introducing the concept of a set of left triangulating idempotents. A criterion for a ring with a complete set of triangulating idempotents to be quasi-Baer is provided. A structure theorem for a quasi-Baer ring with a complete set of triangulating idempotents is shown...

A module M is called (strongly) FI-extending if every fully invariant submodule of M is essential in a (fully invariant) direct summand of M. A ring R with unity is called quasi-Baer if the right annihilator of every ideal is generated, as a right ideal, by an idempotent. For semi-prime rings the FI-extending condition, strongly FI-extending condit...

This paper considers the inheritance of certain properties by subnear-rings and supernear-rings, with the focus being on prime ideals, 3-prime ideals, and various radicals. Playing a key role in this is a common nonzero ideal shared by the near-ring and its subnear-ring. Sharper results are obtained when this common ideal is essential.

Let be a unital K-algebra, where K is a commutative ring with unity. An idempotent is {\it left semicentral\/} if , and is {\it SCI-generated\/} if it is generated as a K-module by left semicentral idempotents. This paper develops the basic properties of SCI-generated algebras and characterizes those that are also prime, semiprime, primitive, o...

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-ext...

If ? is a radical of near-rings and ? is its supplementing radical, then ?(N)??(N) ? N. We address the issue when ?(N) ??(N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N ? Nc is a hereditary Kurosh–Amitsur radical, c is characterized in terms of distributors and criteria are given for the decom...

The main purpose of this paper is to extend the study of various annihilator conditions on polynomials to formal power series in which addition and substitution are used as operations. This process is not as routine as in the ring case because the substitution of one formal power series into another may not be well defined. Two approaches are intro...

We extend a theorem of Kist for commutative PP rings to principally quasi-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Also decompositions of quasi-Baer and principally quasi-Baer rings are investigated.

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 (includin...

In this paper, we initiate the study, in nearrings, of various annihilator conditions which were used by Rickart and Kaplansky to abstract the algebraic properties of Von Neumann algebras. In our main results we were able to extend a result of Armendariz on the Baer condition in a polynomial ring to a Baer condition in a nearring of polynomials. Th...

A (near-) ring R is called left self-distributive, LSD, if vxy = vxvy for all v,x,y in R. Right self-distributive (near-) rings, RSD, are defined similarly. A (near-) ring is called self-distributive, SD, if it is both LSD and RSD. Observe that the class of LSD (left near-) rings is exactly the class of (left near-) rings for which each left multip...

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 re...

This volume is the Proceedings of the Third Korea-China-Japan Inter­ national Symposium on Ring Theory held jointly with the Second Korea­ Japan Joint Ring Theory Seminar which took place at the historical resort area of Korea, Kyongju, June 28-July 3, 1999. It also includes articles by some invited mathematicians who were unable to attend the conf...

In this paper we develop the theory of generalized triangular matrix representation in an abstract setting. This is accomplished by introducing the concept of a set of left triangulating idempotents. These idempotents determine a generalized triangular matrix representation for an algebra. The existence of a set of left triangulating idempotents do...

A module M iscalled a CS-module or an extendingmodule if every submodule is essentialin a direct summand of M. A ring R is called a rightCS-ring or a right extendingring if R_R is a CS-module.For several types of right CS-rings it is known that either all right ideals or some large class ofright ideals inherit the CS property. For example, by a res...

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 tha...

A ring R with unity is called a (quasi-) Baer ring if the left annihilator of every (left ideal) nonempty subset of R is generated (as a left ideal) by an idempotent. Armendariz has shown that if R is a reduced PI-ring whose centre is Baer, then R is Baer. We generalise his result by considering the broader question: when does the (quasi-) Baer con...

A ring R is called a right principally quasi-Baer (or simply right p.q.-Baer) ring if the right annihilator of a lprincipal right ideal is generated by an idempotent. Weshow that a ring R is right p.q.-Baer if and only if R[x]is right p.q.-Baer. This result allow us to generalize results of E. P. Armendariz and S. J

This paper is primarily a survey of the results on essential covers of radical classes of rings. Some problems are posed, and some new results are presented.

In this paper we investigate connections between the condition that every prime ideal is maximal and various generalizations of von Neumann regularity. As a corollary of our results we show that if N is a reduced zerosymmetric right near-ring, then every prime ideal is maximal if and only if N is left weakly regular (i.e., x∈〈x〉x, for all x∈N, wher...

Let P be a prime ideal of a ring R, O(P) = {a R | aRs = 0, for some s R/P} | and Ō(P) = {x R | xn O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this...