In this talk, we will present some characterizations of -semiperfect and -perfect rings in terms of locally (finitely, quasi-, direct-) projective -covers and flat -covers. Also, we investigate some properties of flat modules having a projective -cover.
HACETTEPE UNIVERSITY, ANKARA, TURKEY;
CENTER OF RING THEORY AND ITS APPLICATIONS, OHIO UNIVERSITY,
paydogdu at hacettepe dot edu dot tr
In studying the minimal prime spectra of commutative rings with identity, we have been able to identify several interesting types of extensions of rings, namely, -extension, rigid extensions, -extension, and -extension. In this talk, we will introduce these extensions for commutative rings and establish some relationship between them. In particular, we will determine what kind of ring extensions will result in a homeomorphism of the topologies of the minimal prime spectrum.
PENN STATE ERIE - THE BEHREND COLLEGE
pxb39 at psu dot edu
It all starts with Fitting's Lemma. Some roads it follows.
UNIVERSITY OF IOWA
IOWA CITY, IA
camillo at math dot uiowa dot edu
In this talk I will explain the equivalence between -linear Kerdock codes and certain relative difference sets in Galois rings over .
WRIGHT STATE UNIVERSITY
yuqing dot chen at wright dot edu
Let be a non-empty class of modules closed under isomorphic copies. We consider some classes of modules associated to . Among them, we study two important classes in the theory of natural and conatural classes of modules, namely the class consisting of all modules having no non-zero submodule in , as well as its dual. (joint work with Septimiu Crivei)
crivei at math dot utcluj dot ro
I will look into notions that are in some sense dual to the notion of slenderness. I will also address some fallacies regarding some of the dual notions that have persisted for many years.
rdimitric at juno dot com
We discuss the structure of negacyclic codes of length over the Galois rings . It will be shown that the ambient ring is a chain ring, whose ideals are such negacyclic codes. This structure are then used to completely determined the Hamming and homogeneous distances of all such codes. The technique is then generalized to obtain the structure and Hamming and homogeneous distances of all -constacyclic codes of length over , where is any unit of the ring that has the form , for integers . Among other results, duals of such -constacyclic codes are studied, and necessary and sufficient conditions for the existence of a self-dual -constacyclic code are established.
KENT STATE UNIVERSITY
hdinh at kent dot edu
A module is called a V-module if every simple module in the category is -injective. In their recent paper, Holston, Jain and Leroy introduced the concept of WV-rings: A ring is called a right WV-ring if every cyclic right -module is either isomorphic to or it is a -module. In this talk, we show that if is a right WV-ring which is not right V, then has exactly 3 distinct right ideals such that is a division ring, and .
This is a joint work with C. Holston.
DEPARTMENT OF MATHEMATICS, OHIO UNIVERSITY
huynh at math dot ohiou dot edu
A ring is called a Zassenhaus ring if any homomorphism of the additive group of that leaves all left ideals of invariant, is a left multiplication by some element a of , i dot e. for all . Let be a -bimodule. Then the direct sum turns naturally into a ring by defining . This ring is called the idealization of the module , which is an ideal of . We will investigate conditions under which is a Zassenhaus ring.
Manfred_Dugas at baylor dot edu
A ring is called left pure semisimple if every left -module is a direct sum of finitely generated modules. In 2007, L. Angeleri Hügel introduced and studied a key module in -mod, where is hereditary left pure semisimple and is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective modules in -mod. The module appears to play a significant role in understanding how far the left pure semisimple ring is from being of finite representation type. In this talk we provide an answer to a question by Angeleri Hügel, showing that if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in -mod, then the key module is endofinite if and only if the ring has finite representation type. (This is joint work with José Luis García from the University of Murcia, Spain)
OHIO UNIVERSITY, ZANESVILLE CAMPUS
nguyend2 at ohio dot edu
The subject of isomorphism theorems began with the Baer-Kaplansky Theorem, which states that two abelian groups are isomorphic if and only if there exists a ring isomorphism between their respective endomorphism rings. Subsequent research by Wolfson, May and others has shown that several classes of modules over a discrete valuation domain also possess similar isomorphism theorems. I will show that in many classes of modules over a complete discrete valuation domain, an algebra isomorphism between only the Jacobson radical of the endomorphism rings of two modules is sufficient to imply that the modules are isomorphic.
UNIVERSITY OF HOUSTON
mflagg at math dot uh dot edu
All modules are over an integral domain . An -module is said to be weak-injective (S.B. Lee) if for all -modules of weak-dimension at most 1. (This generalizes Enochs-cotorsion modules which are defined in the same way by using flat modules .) We survey some of the recent results on weak-injective modules.
NEW ORLEANS, LA
fuchs at tulane dot edu
We show how to construct a theory of modules over general rings (i dot e., rings which do not necessarily have an identity element) so that most of the known basic results of the theory of modules for unital rings are obtained by particularizing the results of the general theory. (Joint work with L. Marin)
UNIVERSITY OF MURCIA
jlgarcia at um dot es
This is a summary of results of a paper with Jain and Leroy. is called a right -ring if each simple right -module is injective relative to proper cyclics. If is a right -ring, then is right uniform or a right -ring. It is shown that for a right -ring , is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module with projective socle over a -ring such that every subfactor of is a direct sum of a projective module and a CS or noetherian module, we show , where is semisimple and is noetherian with zero socle. In the case that , we get , where is a semisimple artinian ring, and is a direct sum of right noetherian simple rings with zero socle. In addition, if is a von Neumann regular ring, then it is semisimple artinian.
DEPARTMENT OF MATHEMATICS, OHIO UNIVERSITY
holston at math dot ohiou dot edu
It is known that the polynomial ring is never clean. Among other things, clean elements of and are described.
OHIO UNIVERSITY, ZANESVILLE
pkanwar at ohiou dot edu
Throught this paper all rings considered are associative with identity and all modules are unitary. We investigate in this paper the von Neumann regularity of rings whose essential maximal right ideals are GP-injective. We prove that a ring R is strongly regular if and only if R is a 2-primal ring whose essential maximal right ideals are GP-injective. It is also shown that a ring R is strongly regular if and only if R is a strongly right bounded ring whose essential maximal right ideals are GP-injective. Moreover we show that if R is a PI-ring whose esstial maximal right ideals are GP-injective, then R is strongly phi-regular.(This is a joint work with N.K.Kim and S.B.Nam)
KYUNG HEE UNIVERSITY
YONGIN-SI, GYEONGGI-DO, SOUTH KOREA
jykim at khu dot ac dot kr
One says that the Krull-Schmidt Theorem holds for a class of modules if each module in the class can be decomposed uniquely into a direct sum of indecomposable modules in the class (up to order and isomorphism class of the summands). For most rings, even restricted to the class of finitely generated modules, this fails. In joint work with Basak Ay, we examine this question for commutative rings R and the category of R-modules which are isomorphic to finite direct sums of ideals of R.
FLORIDA ATLANTIC UNIVERSITY
BOCA RATON, FL
klingler at fau dot edu
An equation of the form in called a unit metro equation in a ring . Whenever such an equation holds, we say that is ``completable'', and is ``reflectable''. The study of such elements constitutes an interesting part of the understanding of the noncommutative nature of the ring . In this talk, we'll focus on a prototypical case; namely, where is a matrix ring over a commutative ring . In this case, the completable and reflectable elements of are closely tied to the ring arithmetic of . This study has led to the discovery of some determinantal formulas for computing . A ``trace version'' and a ``supertrace version'' turn out to be special cases of a general ``quantum-trace determinantal formula'' for computing . (This is a joint work with D. Khurana and N. Shomron.)
UNIVERSITY OF CALIFORNIA
lam at math dot berkeley dot edu
Let be any ring with unity and be a unital right -module. Set . In 2004, Rizvi and Roman introduced the notion of a Baer module using the endomorphism ring of the module. is said to be Baer if the right annihilator in of any subset of is generated by an idempotent in . Recently, we extended this notion to one that generalizes the notion of a Baer module as well as that of a p dot p.-ring: is said to be Rickart if the right annihilator in of any single element of is generated by an idempotent .
In this talk, we introduce a notion dual to that of a Rickart module. Namely, is called a dual Rickart module if the image in of any single element of is generated by an idempotent of . We present results and properties of dual Rickart modules and related concepts. We explore conditions which allow for direct sums of dual Rickart modules to be dual Rickart. In particular, we will discuss characterizations of classes of rings using the dual Rickart property of modules over them.
(This is a joint work with S. Tariq Rizvi and Cosmin Roman.)
THE OHIO STATE UNIVERSITY
lgy999 at math dot osu dot edu
We will show that pseudo-linear transformations naturally appear in the context of a module of an Ore extension and give some of their properties. We will then briefly describe their relations with polynomial maps and factorizations in Ore extensions In particular, we will show how to translate factorization of skew polynomials over a finite field twisted by the Frobenius automorphism in terms of factorization of the usual polynomial ring over a finite field.
LENS, PAS DE CALAIS, FRANCE
andreleroy55 at gmail dot com
A module M is said to be poor if it is N-injective only when the module N is semisimple. This talk reports on results about Poor Modules and related notions obtained in recent collaborations with Adel Alahmadi, Mustafa Alkan, Noyan Er, and Nurhan Sokmez.
Among other results, we will show that all rings have poor modules and will precisely characterize those rings that have semisimple poor modules. We also address the question: over which rings does each module have either maximal or minimal domain of injectivity (i dot e. when is each module either poor or injective)? We will present a theorem classifying those rings (to which we refer as "rings without a middle class") and offer various components of a potential converse.
Addressing these questions will highlight how the notion of poor modules and related concepts relate to other widely studied families of rings and modules such as SI-rings, PCI domains, QI rings, etc.
lopez at ohio dot edu
We introduce a hierarchy of notions about algebras having a basis consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras. The other conditions considered in the said hierarchy include the requirement that for an invertible basis , the set of inverses be itself a basis, the notion that be closed under inverses and the idea that be closed under products. It is shown that the last property is unique of group rings. Many examples are considered and it is determined that the hierarchy is for the most part strict. For any ﬁeld , all semisimple -algebras are invertible. Semisimple invertible -algebras are fully characterized. Likewise, the question of which single-variable polynomials over a ﬁeld yield invertible quotient rings of the -algebra is completely answered. Connections between invertible algebras and -rings (rings generated by units) are also explored.
moore at math dot ohiou dot edu
A module is called a lifting module if, any submodule of > contains a direct summand of such that is small in . It is known that a finite direct sum of lifting modules need not be lifting. It was investigated this question by using (*)-generalized projective modules (joint work with Ummuhan Acar).
CENTER OF RING THEORY AND ITS APPLICATIONS, OHIO UNIVERSITY
orhannil at yahoo dot com
This is currently in a preliminary state. I will send an abstract after additional work.
HIGHLAND PARK, NJ
osofskyb at member dot ams dot org
A ring is called quasi-Baer if the right annihilator of every ideal of is generated by an idempotent. The quasi-Baer condition of is discussed when is a quasi-Baer ring, where is the prime radical of . We provide an example of quasi-Baer ring such that is not quasi-Baer. However, when is nilpotent, it is shown that if is a quasi-Baer (resp., Baer) ring, then is quasi-Baer (resp., Baer). Examples which illustrate and delimit the results of this paper are also discussed (this is a joint work with Gary F. Birkenmeier and Jin Yong Kim).
DEPARTMENT OF MATHEMATICS, BUSAN NATIONAL UNIVERSITY
BUSAN 609-735, SOUTH KOREA
jkpark at pusan dot ac dot kr
In earlier work with T.Y. Lam, we unified several results from commutative algebra of the form "maximal implies prime" through the "Prime Ideal Principle." After a brief review of the commutative case, we will discuss a generalization of this method to noncommutative rings. This requires us to define a new type of "prime right ideal." We will illustrate these new ideas with several examples and applications.
UNIVERSITY OF CALIFORNIA, BERKELEY
mreyes at math dot berkeley dot edu
Let be any ring and be an -module. Set . In 2004, we introduced the notion of a Baer module, generalizing the ring theoretic concept of Baer rings. is said to be Baer if the right annihilator in of any subset of is generated by an idempotent in . Equivalently, the left annihilator in of any submodule of is generated by an idempotent in . We recently extended this notion to that of a Rickart module, bringing the ring theoretic concept of Rickart (or PP-) rings to module theoretic setting. is called a Rickart module if the right annihilator in of any single element of is generated by an idempotent in , equivalently, for every .
In this talk we present results and properties of Rickart modules and related concepts. It is known that the direct sum of Rickart modules is not Rickart in general. We explore conditions which allow for direct sums of Rickart modules to be Rickart. In particular, we investigate when free and projective -modules have the Rickart property and characterize the class of rings for which this happens.
(This is a joint work with G. Lee and S. T. Rizvi)
THE OHIO STATE UNIVERSITY, LIMA
cosmin at math dot ohio-state dot edu
Let be a commutative ring with identity and let be an infinite unitary module over . We call homomorphically congruent (HC for short) provided for every submodule of for which . We classify HC modules over Dedekind domains, extending Scott's classification over , and give other results over arbitrary commutative rings.
UNIVERSITY OF EVANSVILLE
as341 at evansville dot edu
For a field we consider systems consisting of a finite dimensional -vector space , a nilpotent -linear operator and two subspaces , of such that .
Each system is a direct sum of indecomposable ones; the complexity of the classification problem for the indecomposable systems increases with the nilpotency index of . For , there are only finitely many indecomposable systems, up to isomorphy, while for the classification problem is considered infeasible.
In this talk we discuss vector spaces with two invariant subspaces in the critical case where . In particular, we present a full list of the dimension types of the indecomposable systems. This is a report on joint work with Audrey Moore (Delaware State University).
FLORIDA ATLANTIC UNIVERSITY
BOCA RATON, FL
markus at math dot fau dot edu
An element in a ring is called potent (respectively, -potent) if it equals one of its proper powers (respectively its -th power), and periodic if two of its powers are equal. A (commutative) ring in which each element is -potent, potent, or periodic is called respectively, -Boolean, potent, or periodic. The classical example are Boolean rings (= -Boolean in our terminology), and Stone proved that a ring is Boolean if and only if it embeds in a power set ring. I will give similar characterizations for the more general rings discussed here. As a corollary, I obtain that a ring is periodic if and only if every finitely generated subring is finite.
CITY UNIVERSITY OF NEW YORK
NEW YORK CITY, NY
hschoutens at citytech dot cuny dot edu
Call a nonzero R-module "torsion" if all of its elements have nonzero annihilator in R, and "faithful" when the module has zero annihilator in R. Rings admitting torsion modules are completely classified as non-division rings. What rings admit faithful torsion modules? What is the minimum size of a generating set for a faithful torsion module of such a ring? These questions are addressed for several classes of rings, including quasi-Frobenius, unit regular, duo serial, trivial socle, and commutative semiprimitive rings. (Joint work with Greg Oman)
schwiebert at math dot ohiou dot edu
A right ideal I of a ring R is called small if, for any right ideal K of R, I+K=R informs K=R. A right R-module M is called small injective if every homomorphism from a small right ideal of R to M can be extended to one from R to M. Some results on small injectivity are given and some known results are improved.
lshen at seu dot edu dot cn
A module is called -injective if every direct sum of copies of is injective. We will present some new characterizations of -injective modules. We will also discuss some applications of the study of -injective modules.
ST. LOUIS UNIVERSITY
ST. LOUIS, MO
asrivas3 at slu dot edu