Theory of Rings and Modules Abstracts

HULLS FOR RINGS AND MODULES FROM CERTAIN CLASSES

GARY BIRKENMEIER, University of Louisiana at Lafayette, Lafayette, LA

gfb1127 at louisiana.edu

Abstract

In this talk we consider results and examples of hulls of rings and modules from certain classes including the existence and uniqueness of such hulls.  Also we will consider the problem of the termination of an ascending sequence of right essential overrings

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ON THE QUASI-STRATIFIED ALGEBRAS OF LIU

AND PAQUETTE

WALTER BURGESS, University of Ottawa, Canada

wburgess at uottawa.ca

(joint with A Mojiri)

Abstract

These interesting algebras have the property that the Cartan determinant conjecture and its converse hold for them.  This talk will compare these algebras with left serial algebras (which also have the property), will show that the Yamagata construction yields quasi-stratified algebras and show other constructions of them.

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A GENERALIZATION OF THE DUAL ISOMORPHISM

THEOREM

VICTOR CAMILLO, University of Iowa, Iowa City, IA

camillo at math.uiowa.edu

Abstract

We discuss consequences of the dual isomorphism theorem and raise some questions.

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RELATIVE FLATNESS AND PURITY

JOHN DAUNS, Tulane University, New Orleans, LA

dauns at tulane.edu

Abstract

For a right $R$ -module $M$ , let $\sigma [M]$ be the subcategory of Mod-$R$ subgenerated by $M$ . A left $R$ -module $F$ $\in$ $R$ -Mod is   $\sigma [M]$ - flat, if for any $X<Y\in \sigma [M]$ , $0\longrightarrow X\otimes F \longrightarrow Y\otimes F$ remains exact. The module $F$ is TM-flat, if for any $X\in \sigma [M]$ , Tor $_1^R(X,F)=0$ .  A short exact sequence of left $R$ -modules $0\longrightarrow A \longrightarrow B$ is $\sigma [M]$ - pure if for any $X\in \sigma [M]$ ,  $0\longrightarrow X\otimes A \longrightarrow X\otimes B$ remains exact.

The theory of flatness and purity relative to the category $\sigma [M]$ is developed, including the following. A module ${}_R F$ is $\sigma [M]$ -flat $\Longleftrightarrow$ $F^*_R=$ Hom$_Z(F,Q/Z)$ is $M$ -injective ( i.e.,  $\sigma [M]$ -injective). For a short exact sequence of left $R$ -modules ${\mathcal E}$ :  $0\longrightarrow A \longrightarrow B\longrightarrow C\longrightarrow 0$ the connection between the following two properties (a) and (b) is determined. (a) $C$ $\sigma [M]$ - flat (resp. TM-flat).   (b) ${\mathcal E}$ is  $\sigma [M]$ - pure.

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COMPLETE DISTANCES OF NEGACYCLIC CODES

OF LENGTH $2^s$ OVER $\mathbb Z_{2^a}$

HAI DINH, Kent State University, Kent, OH

hdinh at kent.edu

Abstract

It was shown that negacyclic codes of length $2^s$ over $\mathbb{Z}_{2^a}$ are precisely the ideals of the chain ring $\frac{\mathbb{Z}_{2^a}[x]}{\langle x^{2^s}+1 \rangle}$ . Using this ring theoritic chain structure, various kinds of distances of all negayclic codes of length $2^s$ over $\mathbb{Z}_{2^a}$ are completely determined. We first calculate the Hamming distances of all such negacyclic codes, which particlularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes over the ring $\mathbb{Z}_{2^a}$ , namely, the $\lambda$ -constacyclic codes of length $2^s$ over $\mathbb{Z}_{2^a}$ , where $\lambda$ is any unit of $\mathbb{Z}_{2^a}$ with the form $4k-1$ . We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes.

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SOME RESULTS ON SIMPLE RINGS

DINH VAN HUYNH, Ohio University, Athens, OH

huynh at math.ohiou.edu

(joint work with John Clark)

Abstract

Using a variation on the concept of a CS module, we describe exactly when a simple ring is isomorphic to a ring of matrices over a Bézout domain. Our techniques are then applied to characterize simple rings which are either right and left Goldie, right and left semihereditary, or right self-injective.

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DESCRIPTION OF THE INJECTIVE MODULES OVER

THE FIRST WEYL ALGEBRA

ALINA DUCA, University of Manitoba, Canada

umducaan at cc.umanitoba.ca

Abstract

A classical result due to E.Matlis says that over a left Noetherian ring every injective left module is a direct sum of indecomposable injective left modules. It follows that over the first Weyl algebra $Weyl$ every indecomposable injective is the injective envelope $E=E(M)$ of a simple module $M$ . Since an arbitrary simple module $M$ is $k[x]$ or $k[y]$ -torsion-free, the ring $Weyl$ can be localized at $k[x]\setminus \{0\}$ or at $k[y] \setminus\{0\}$ to a principal left (and right) ideal domain $R$ .

Motivated by a classic treatment of O.Ore, we take advantage of some uniqueness decomposition theorems in $R$ and present a nice description of the internal structure of the indecomposable $E$ . Furthermore, a consideration of the $R$ -action on $E$ and of the arithmetic of $R$ on an element-by-element basis leads to a new localization of $Weyl$ (a subring of the Weyl division algebra). Consequently, we can investigate even further the structure of $E$ and determine its socle series. In addition, this extension of $Weyl$ allows us to pursue a detailed analysis of the structure of the endomorphism ring and that of the bicommutator of $E$ .

[O.Ore, Theory of non-commutative polynomials., "Annals of Math.", 1932, vol.34, 480-508]

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ENDOPROPERTIES OF MODULES AND PURE

SEMISIMPLE RINGS

NGUYEN VIET DUNG, Ohio University, Zanesville, OH

nguyend2 at ohio.edu

(joint work with Jose Luis Garcia)

Abstract

It is a well-known result, due to M. Prest and B. Huisgen-Zimmermann and W. Zimmermann, that a ring $R$ is of finite representation type if and only if every right $R$ -module is of finite length over its endomorphism ring. A ring $R$ is called left pure semisimple if every left $R$ -module is a direct sum of finitely generated modules. Left and right pure semisimple rings are precisely rings of finite representation type, but it is still unknown whether left pure semisimple rings are always right pure semisimple. In this talk, we discuss a number of characterizations of pure semisimple rings and rings of finite representation type, in terms of endoproperties of their modules, extending the above mentioned result. Our results are closely related with restricted versions of the pure semisimplicity conjecture.

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THE ABSOLUTE ARITHMETIC CONTINUUM AND

THE UNIFICATION OF ALL NUMBERS GREAT AND SMALL

PHILIP EHRLICH, Ohio University, Athens, OH

ehrlich at ohiou.edu

Abstract

In his monograph/ On Numbers and Games/ [1976], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including -w, w/2, 1/w, ÷w and w-p to name only a few. Indeed, this particular real-closed field, which Conway calls/ No/, is so remarkably inclusive that, subject to the proviso that numbers-construed here as members of ordered "number" fields-be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG, it may be said to contain "All Numbers Great and Small." In this respect,/ No/ bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields.        However, in addition to its distinguished structure as an ordered field,/ No/ has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or/ simplicity hierarchy/, as we have called it [1994], depends upon/ No/'s (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with/ No/'s structure as an ordered group and an ordered field, respectively, it being understood that/ x/ is/ simpler than y/ just in case/ x/ is a predecessor of/ y/ in the tree.     In a number of earlier works [Ehrlich 1987; 1989; 1992], we suggested that whereas the real number system should merely be regarded as constituting an Archimedean arithmetic continuum, the system of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In this paper, we will outline some of the properties of the system of surreal numbers that emerged in [Ehrlich 1988; 1992; 1994; 2001] which lend credence to this thesis, and draw attention to some important respects in which the theory of surreal numbers may be regarded as vast generalization of Cantor's theory of ordinals, a generalization which also provides a setting for Abraham Robinson's [1961] infinitesimal approach to analysis as well as for the profound and largely overlooked non-Cantorian theories of the infinite (and infinitesimal) pioneered by Giuseppe Veronese [1891], Tullio Levi-Civita [1892; 1898], David Hilbert [1899] and Hans Hahn [1907] in connection with their work on non-Archimedean ordered algebraic and geometric systems and by Paul du Bois-Reymond [1870-71; 1882], Otto Stolz [1883], G. H. Hardy [1910; 1912] and Felix Hausdorff [1909] in connection with their work on the rate of growth of real functions.

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INDECOMPOSABLE FLAT COTORSION MODULES

PEDRO GUIL ASENSIO, University of Murcia, Spain

paguil at um.es

Abstract

We develop a method for constructing indecomposable flat cotorsion modules in the category of flat modules over a ring R. This allows to show that the set of these modules is a "cogenerator" of this category. Our methods are inspired by Auslander's work on Representation Theory.

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HYPERCENTRAL UNITS IN ALTERNATIVE LOOP RINGS

ANJANA KAILA, Punjabi University, India

an14in at yahoo.com

(joint work with A. K. Bhandari)

Abstract

An alternative ring is a ring (not necessarily associative) in which $x(xy)=x^{2}y$ and $(xy)y=xy^{2}$ are identities. A Moufang loop is a loop in which $x(y.xz)=(xy.x)z$ is an identity. An RA (ring alternative) loop is a Moufang loop $L$ whose loop ring $RL$ , for a commutative and associative ring $R$ of characteristic different from 2, is alternative but not associative. We show that if $L$ is not a Hamiltonian Moufang 2-loop, then ${\cal U}_{1}(\mathbb{Z}L)$ , the loop of normalized units of integral loop ring for any RA loop $L$ is of central height 1. In case $L$ is a Hamiltonian Moufang 2-loop, obviously, ${\cal U}_{1}(\mathbb{Z}L)=L$ is of central height 2.

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SEMILOCAL CONTINUOUS GROUP ALGEBRAS

PRAMOD KANWAR, Ohio University, Zanesville, OH

pkanwar at math.ohiou.edu

Abstract

A ring $R$ is said to be right CS if every right ideal is essential in a summand of $R$ . A right CS ring is said to be right continuous if any right ideal of $R$ which is isomorphic to a summand of $R$ is itself a summand of $% R$ . Right CS rings and right continuous rings are generalizations of right selfinjective rings. It is known that the group algebra $KG$ of a group $G$ over a field $K$ is selfinjective if and only if $G$ is a finite group. We discuss group algebras that are continuous and give the corresponding conditions on the group $G$ . Among others, it is shown that (i) a semilocal group algebra $KG$ of an infinite nilpotent group $G$ over a field $K$ of characteristic $p>0$ is CS (equivalently, continuous) if and only if $% G=P\times H$ , where $P$ is an infinite locally finite $p$ -group and $H$ is a finite abelian group whose order is not divisible by $p$ , (ii) if $K$ is a field of characteristic $p>0$ and $G=P\times H$ where $P$ is an infinite locally finite $p$ -group (not necessarily nilpotent) and $H$ is a finite group whose order is not divisible by $p$ then $KG$ is CS if and only if $H$ is abelian. Furthermore, a commutative semilocal group algebra is always continuous.

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CLEAN RINGS AND MODULES

DINESH KHURANA, Panjab University, India

dkhurana at pu.ac.in

Abstract

An element of a ring is called clean if it is a sum of a unit and an idempotent. A ring with all elements clean is called clean ring and a module whose endomorphism ring is clean is called a clean module. We will present some recent developements in the theory of clean rings and modules.

in

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ROW-FINITE MATRICES AND POWER SERIES

NEAR-RINGS

ENOCHS LEE, Auburn University, Montgomery, AL

elee4 at mail.aum.edu

Abstract

We define and investigate (structural) row-finite matrices, lower triangular matrices, and power series in near-rings by using inverse limits of some special classes of near-ring matrices. We show polynomials can be embedded in power series and power series can be embedded in lower triangular matrices as in rings.  A natural topology is defined on lower triangular matrices by generalizing the concept of order of power series.

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ON ANNELIDAN, BÉZOUT AND DISTRIBUTIVE RINGS

GREG MARKS, St. Louis University, St. Louis, MO

marks at slu.edu

(joint work with Ryszard Mazurek)

Abstract

A ring is called right annelidan if the right annihilator of any element of the ring is a right waist, i.e. comparable with every other right ideal.  Right annelidan rings have a nice characterization within the class of right Bézout rings and within the class of right distributive rings (i.e. rings with a distributive lattice of right ideals).  The Bézout or distributive condition enables one to realize a right annelidan ring as a right order in a right uniserial ring.  For Bézout or distributive rings with any of a number of mild chain conditions, the annelidan condition is left-right symmetric.

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NEAT BÉZOUT DOMAINS ARE ELEMENTARY DIVISOR

DOMAINS

WARREN MCGOVERN, Bowling Green State University, Bowling Green, OH

warrenb at bgnet.bgsu.edu

Abstract

Throughout $A$ denotes a commutative ring with identity. $A$ is called clean if every element can be written as the sum of a unit and an idempotent. If every proper homomorphic image is clean then $A$ is called neat.

We will discuss the class of neat Bézout domains and some of its generalizations. In particular, every such domain is an elementary divisor domain.

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OSOFSKY COMPATIBLE RINGS

JAE KEOL PARK, Busan National University, South Korea

jkpark at pusan.ac.kr

(joint work with G.F. BIRKENMEIER and S.T. RIZVI)

Abstract

A ring $R$ is called right Osofsky compatible if a right injective hull $E$ of $R$ has a ring multiplication which extends the $R$ -module scalar multiplication of $E$ over $R$ . Right nonsingular rings are right Osofsky compatible. A class of right Osofsky compatible rings with $R = Q(R)$ will be discussed, where $Q(R)$ is a maximal right ring of quotients of $R$ .

Also we discuss a class of rings $R$ such that $R = Q(R)$ and there exists a right essential $R$ -module extension $S$ which satisfies:

(i) there are three ring structures $S(1)$ , $S(2)$ , and $S(3)$ on $S$ whose ring multiplications     extend the $R$ -module scalar multiplication of $S$ over $R$ .

(ii) $S(1)$ and $S(2)$ are QF, but not ring isomorphic.

(iii) $S(3)$ is not even right FI-extending.

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ON $\mathcal K$ -NONSINGULARITY AND CONTINUOUS MODULES

COSMIN ROMAN, The Ohio State University, Lima, OH

cosmin at math.ohio-state.edu

(joint work with S.T. Rizvi)

Abstract

A module $M$ is $K$ -nonsingular iff $Ker\varphi$ is essential in $M$ implies $\varphi=0$ , for all $\varphi\in End(M)$ . For a module $M$ , nonsingular implies polyform implies $K$ -nonsingular, while reverse implications are not true. We provide internal characterizations of $K$ -nonsingular continuous modules of various types. Our theory properly extends the well-known theory of decomposition of nonsingular injective modules into types, replacing nonsingularity by $K$ -nonsingularity, and injectivity by continuity. Our internal characterizations are analogous to the ones obtained by Goodearl and Boyle for the nonsingular injective modules. As a consequence we obtain a characterization of arbitrary $K$ -nonsingular continuous modules.

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SOME OPEN PROBLEMS ON WEAK-INJECTIVITY

AND WEAK-PROJECTIVITY

MOHAMMAD SALEH, Birzeit University, Palestine

msaleh at birzeit.edu

(joint work with Ali Abdel-Mohsen)

Abstract

The purpose of this paper is to survey some of the most important results on the theory of weakly injective and weakly projective modules a generalization of injective and projective modules and raise some of the fundamental open problems in this area. It is shown that For a module $M$ , there exists a module $K\in \sigma[M]$ such that $K\oplus N$ is weakly injective in $\sigma [M]$ , for any $N\in \sigma[M]$ . Similarly, if $M$ is projective and right perfect in $\sigma [M]$ , then there exists a module $K\in \sigma[M]$ such that $K\oplus N$ is weakly projective in $\sigma [M]$ , for any $N\in \sigma[M]$ . Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. Among others, For some classes ${\mathcal K}$ of modules in $\sigma [M]$ we study when direct sums of modules from ${\mathcal K}$ satisfies a property $I\!\!P$ in $\sigma [M]$ . In particular, we get characterization of locally countably thick modules a generalization of locally $q.f.d.$ modules. Characterizations of rings over which every weakly injective is weakly projective and conversely are given. Finally, we conclude with several open questions. The following are some of these problems:

1. Characterize rings over which every (weakly)tight right R-module is weakly projective (cotight).
2. Characterize rings over which every cotight right R-module is weakly injective (tight, weakly tight).
3. Characterize rings over which the product of weakly projective (cotight) is weakly projective (cotight).

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ORTHOGONAL COVERS OF GRAPHS VIA

MODULE ENDOMORPHISMS

MARKUS SCHMIDMEIER, Florida Atlantic University, Boca Raton, FL

mschmidm at fau.edu

(joint work with Hans-Dietrich Gronau, Rostock University, Germany)

Abstract

Let $K$ be the complete oriented graph on the finite set $I$ . A family $(G_i)_{i\in I}$ of spanning subgraphs is said to be an orthogonal cover if (1) every arrow in $K$ occurs in exactly one of the $G_i$ and (2) for any pair $(i,j)$ , the graph $G_i$ and the opposite of the graph $G_j$ have exactly one arrow in common.

Orthogonal covers are frequently studied in combinatorics. Particular interest is in covers which are polycyclic (each $G_i$ is a disjoint union of cycles) and isophyllic (all the $G_i$ are isomorphic as graphs).

It turns out that endomorphisms of finite modules yield a rich variety of orthogonal covers; suitable choices for the endomorphisms produce covers which are both polycyclic and isophyllic.

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IDEAL THEORY IN LOCAL RINGS OF FINITE

EMBEDDING DIMENSION

HANS SCHOUTENS, City University of New York, New York, NY

hschoutens at citytech.cuny.edu

Abstract

A commutative local ring of finite embedding dimension has a Noetherian completion. This allows us to apply some tools from commutative algebra to this larger class. In particular, we get primary decomposition for closed ideals. If the ring is moreover geometric, that is to say, has the same (Krull) dimension as its completion, then the lattice of ideals is well-behaved. This leads to several Noetherianity criteria for such rings.

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CLASS OF DIAGRAM ALGEBRAS

PARVATHI SIVASUBRAMANIAM, Ramanujan Institute

for Advanced Study in Mathematics, India

sparvathi at hotmail.com

Abstract

Diagram algebras are associative algebras having a basis consisting of diagrams. Partition algebras arose in the context of Potts model in statistical mechanics and as generalization of Temperely Lieb algebras. The diagram algebras include partition algebras, Brauer algebras, Temperely Lieb algebras, planar algebras and the group algebras of the symmetric group.

The study of diagram algebras has proved to be valuble to both ring theorists and physicsts(cf.Paul Martin, Temperely Lieb algebras for non-planar statistical mechanics- the partition algebra construction, J.Knot Theory and Ramifications., Vol 3,No 1 ( 1994), 51-82). These diagram algebras have shown to have strong connection with Lie theory.

A new class of diagram algebras namely Signed Brauer Algebras, G-Brauer algebras and Vertex colored Partition algebra havae been introduced by us. The structure and representation of the above algebras will be discussed and the Schur-Weyl duality will be explained in detail

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RIGHT FINITELY $\Sigma$ -$Q$ RINGS

ASHISH KUMAR SRIVASTAVA, Ohio University, Athens, OH

ashish at math.ohiou.edu

(joint work with S.K. Jain and Surjeet Singh)

Abstract

It is well known that a ring $R$ in which each right ideal is a finite direct sum of injective right ideals is semisimple artinian. We introduce rings in which each right ideal is a finite direct sum of quasi-injective right ideals. Such rings will be called right finitely $\Sigma$ -$q$ rings. These rings are a natural generalization of right $q$ -rings, which are defined as rings in which each right ideal is quasi-injective. We characterize various classes of right finitely $\Sigma$ -$q$ rings and give some examples.

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ON GALOIS ALGEBRAS SATISFYING THE

FUNDAMENTAL THEOREM

GEORGE SZETO, Bradley University, Peoria, IL

szeto at bradley.edu

Abstract

Let $B$ be a Galois algebra over a commutative ring $R$ with Galois group $G$ such that the subalgebra of the elements fixed under the elements of a subgroup is separable.  If $B$ satisfies the fundamental theorem for Galois extensions, then $B$ is one of the following classes: (1)  indecomposable commutative, (2)  a direct sum of $Re$ and $R(1-e)$ for some minimal central idempotents $e$ and $(1-e)$ , and (3)   indecomposable such that the commutator subalgebra of each separable subalgebra is a direct sum of certain projective  submodules of $B$ induced by $G$ .

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WHEN IS A COALGEBRA A GENERATOR?

BLAS TORECILLAS, University of Almeria, Spain

btorreci at ual.es

Abstract

Let $C$ be a coalgebra over a field $k$ . If $C$ is a left quasi-coFrobenius coalgebra, then $C$ is generator for the category of left comodules. We show that the converse is true if $C$ has a finite coradical series.

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CODE EQUIVALENCE AND FINITE FROBENIUS RINGS

JAY WOOD, Western Michigan University, Kalamazoo, MI

jay.wood at wmich.edu

Abstract

Linear codes defined over a finite Frobenius ring have an extension property--any linear isomorphism between codes that preserves Hamming weight necessarily extends to a monomial equivalence.  The converse is also true--if linear codes over a finite ring have the extension property, then the ring is necessarily Frobenius.  This talk will describe the main ideas in the proof of the converse, following a strategy of Dinh and Lopez-Permouth.

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BAER AND QUASI-BAER PROPERTIES OF GROUP RINGS

YIQIANG ZHOU, Memorial University of Newfoundland, Canada

zhou at math.mun.ca

Abstract

A ring R is called Baer (resp. quasi-Baer) if the left annihilator of any nonempty subset (resp. any ideal) of R is generated by an idempotent. It is unclear when a group ring is (quasi-) Baer. In this talk, we present recent progress towards this question.

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