Theory of Rings and Modules Abstracts

Locally Finitely Generated Grothendieck Categories and

Simple Objects

Toma Albu,

S. Stoilow Institute of Mathematics of the Romanian Academy

A Grothendieck category ${\cal C}$ is said to be locally finitely generated if the subobject lattice of every object in ${\cal C}$ is compactly generated, or equivalently, if ${\cal C}$ possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category ${\cal C}$ indecomposable if ${\cal C}$ is not equivalent to a product of nonzero Grothendieck categories ${\cal C}_{1}\times{\cal C}_{2}$ . In this talk an example of an indecomposable non-locally finitely generated Grothendieck category possessing simple objects is presented, answering in the negative a sharper form of a question posed by Albu, Iosif, and Teply in [J. Algebra, 284 (2005), 52-79].

The results which will be presented have been obtained jointly with John van den Berg.

BUCHAREST, ROMANIA
Toma.Albu at imar.ro

On Parallel Sums and Invariance of Harmonic Mean.

Preliminary Report.

Brian Blackwood, Ohio University

The question on the invariance of harmonic mean of two elements in a von Neumann regular ring will be discussed.

DEPARTMENT OF MATHEMATICS
ATHENS, OH
blackwood at math.ohiou.edu

Constacyclic codes of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$

Hai Q. Dinh, Kent State University

All constacyclic codes of length $p^s$ over the finite chain ring $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ are studied. The units of the ring $R$ are of the forms $\gamma$ , and $\alpha + u\beta$ , where $\alpha, \beta, \gamma$ are nonzero elements of the Galois field $\mathbb{F}_{p^m}$ , which provide $p^m(p^m − 1)$ such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length $p^s$ over the finite field $\mathbb{F}_{p^m}$ are obtained, and used as a tool to establish the structure and Hamming distances of all $(\alpha + u\beta)$ -constacyclic codes of length $p^s$ over R. We then classify all $\gamma$ -constacyclic codes of length $p^s$ over $R$ by categorizing them into 4 types: trivial ideals, principal ideals with nonmonic polynomial generators, principal ideals with monic polynomial generators, and nonprincipal ideals; and we give a detailed structure of ideals in each type. Among other results, we are also able to obtain the number of codewords in each constacyclic code.

DEPARTMENT OF MATHEMATICAL SCIENCES
4314 MAHONING AVENUE
WARREN, OH 44485
hdinh at kent.edu

Rings characterized by finendo modules

Nguyen Viet Dung, Ohio University - Zanesville

Following C. Faith, a module $M$ over an associative ring $R$ (with identity) is called finendo if $M$ is finitely generated over its endomorphism ring. In this talk, we discuss rings satisfying the property that every right $R$ -module is finendo, and show that if such a ring $R$ is hereditary, then $R$ is of finite representation type. We also show that if $R$ is an arbitrary ring with all right $R$ -modules finendo, then $R$ is a left pure semisimple ring with a right Morita duality and the quotient ring $R/(J(R))^2$ is of finite representation type. (This is joint work with José Luis García).

1425 NEWARK ROAD
ZANESVILLE, OH 43701
nguyend2 at ohiou.edu

A weaker form of p-injectivity

Dinh Hai Hoang, Mahidol University

Let R be a ring. A right R-module $N$ is called an $M$ -p-injective module if any homomorphism from an $M$ -cyclic submodule of $M$ can be extended to $M$ . In this paper, we introduce and investigate the class of $M$ -rp-injective modules and $M$ -lp-injective modules, and prove that for a finitely generated Kasch module $M$ , if M is quasi-rp-injective, then there is a bijection between the class of maximal submodules of $M$ and the class of minimal left right ideals of its endomorphism ring $S.$ As an application, if the ring $R$ is right Kasch, right self rp-injective, then there is a bijection between the class of maximal right ideals and the class of minimal left ideals.

RAMA 6
BANGKOK 10400
THAILAND
haiedu93 at yahoo.com

Characterizations of some rings with chain conditions

Dinh Van Huynh, Ohio University

A module $M$ is called a CS-module if every submodule of $M$ is essential in a direct summand. We will use this condition and its generalizations to characterize noetherian rings and QF-rings.

MATH DEPT
321 MORTON HALL
ATHENS, OH 45701
huynh at math.ohiou.edu

Lie Regular Units - Preliminary Report

Pramod Kanwar, Ohio University

An element of a ring /R/ is said to be Lie regular if it can be expressed as a Lie product of an idempotent element in /R/ and a unit in /R/. A unit in /R/ is said to be a Lie regular unit if it is Lie regular as an element of R. Among other things we obtain presentation of some linear groups in terms of Lie regular elements. (This is a joint work with R.K.Sharma and Pooja Yadav)

1425 NEWARK RD
ZANESVILLE, OH 43701
pkanwar at math.ohiou.edu

Some characterizations of QF-rings via

injectivity and small injectivity

Thuyet Van Le, Hue University & Ohio University

A ring $R$ is QF if $R$ is a right or left self-injective ring satisfying ACC on right annihilators. A right $R$ -module $M_R$ is called small injective if every homomorphism from a small right ideal to $M_R$ can be extended to a $R$ -homomorphism from $R_R$ to $M_R$ and a ring $R$ is called right small injective, if $R_R$ is small injective. Some characterizations of QF-rings were obtained, e.g., a ring $R$ is QF iff $R$ is right small injective and has a finitely generated essential right socle. We also prove that if every simple right (resp., left) $R$ -module is small injective, then $R$ is semiprimitive. We also have: The Jacobson radical $J$ of a ring $R$ is noetherian as a right $R$ -module iff $E^{(\mathbb{N})}$ is small injective for every small injective module $E_R$ .

DEPARTMENT OF MATHEMATICS
3 LE LOI STREET
HUE, VIETNAM
t.le at math.ohiou.edu

Rickart Modules (Preliminary Report)

Gangyong Lee, The Ohio State University

A module $M_R$ is called a Rickart module if the right annihilator in $M$ of a principal left ideal of $S$ is generated by an idempotent in $S$ . This concept provides a generalization of a right PP ring to the general module theoretic setting. It is clear that every Baer module (and ring) is Rickart module while the converse is not true. For example, $\mathbb{Z}^{(\mathbb{N})}$ is Rickart but not Baer as a $\mathbb{Z}$ -module. We will obtain characterizations of Rickart modules and discuss various properties. In particular connections between a Rickart module and its endomorphism ring will be presented. For example, $M$ is a Rickart module iff $S$ is a right Rickart ring and $M$ is principal-retractable.

(This is joint work with S. Tariq Rizvi and Cosmin Roman.)

DEPARTMENT OF MATHEMATICS
231 W 18TH AVE
COLUMBUS, OH 43210
lgy999 at math.osu.edu

Connections between various generalizations of the concept of

cyclicity of codes

Sergio Roberto Lopez-Permouth, Ohio University

Cyclic codes play a central role in Coding Theory. Some of the most important codes, such as Reed-Solomon and BCH codes for example, are cyclic. For this reason, generalizations of the concept of cyclicity have surfaced frequently in the literature. One such a generalization, the concept of a consta-cyclic code extends further in a natural way to the concept of a polynomial code. Recently, our group has considered a different type of generalization, the so-called sequential codes. In this talk we explore some surprising connections between these various generalizations. (This talk is based on an ongoing collaboration with Benigno Parra-Avila and Steve Szabo.)

321 MORTON HALL
ATHENS, OH 45701
lopez at ohio.edu

Minimal extensions of rings

Zak Mesyan, USC

A ring S is said to be a minimal extension of a ring R if R is a subring of S and there are no subrings strictly between R and S. I will discuss minimal extensions of an arbitrary ring R, with particular focus on those possessing a nonzero ideal that intersects R trivially. I will also give a classification of all minimal extensions of prime rings. This is joint work with Tom Dorsey, and it generalizes results of Dobbs, Dobbs-Shapiro, and Ferrand-Olivier for commutative rings.

3620 VERMONT AVENUE, KAP 108
LOS ANGELES, CA 90089
mesyan at usc.edu

Quasi-Baer Ring Hulls and Factor Rings of

Quasi-Baer Rings by Prime Radicals

Jae Keol Park, Busan National University, South Korea

We discuss quasi-Baer and FI-extending ring hulls. When a ring $R$ is semiprime, we show that the existence of the quasi-Baer ring hull and the FI-extending ring hull. Also we establish their structures for a certain semiprime ring. Further, FI-extending module hulls are discussed for finitely generated projective modules over semiprime rings. Applications to $C^*$ -algebras are considered.

The quasi-Baer condition of $R/P(R)$ is provided when $R$ is a quasi-Baer ring, where $P(R)$ is the prime radical of $R$ . We give an example of a quasi-Baer ring $R$ such that $R/P(R)$ is not quasi-Baer. 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 illustrate and delimit results are also discussed. (These are joint works with Gary F. Birkenmeier, Jin Yong Kim, and S. Tariq Rizvi).

DEPARTMENT OF MATHEMATICS
BUSAN 609-735
SOUTH KOREA
jkpark at pusan.ac.kr

Rational Power Series and Periodicity of Sequences

Benigno R Parra, Ohio University

It is quite familiar that the real numbers whose decimal expansion is periodic are precisely the rational numbers. If we assume that $R$ is a commutative ring, it is also straightforward to see that a power series $f$ in $R[[x]]$ with periodic coefficients is rational. Now an immediate question is when the converse holds. In this talk we show that periodicity and rationality are equivalent if and only if $R$ is an integral extension of $Z_m$ .. If $F$ is a field, then we also prove the equivalence between two versions of rationality in $F[[x_1,\dots, x_n]]$ . Finally we extend Kronecker's criterion for rationality in $F[[x]]$ to $F[[x_1,\dots, x_n]]$ .

167 1/2 MORRIS AVE APT 1
ATHENS, OH 45701
parra at math.ohiou.edu

TBA

Cosmin Roman, The Ohio State University

TBA

GALVIN HALL
4240 CAMPUS DR
LIMA, OH 45804
cosmin at math.ohio-state.edu

Definable subcategories of modules

Philipp S Rothmaler, CUNY

Full subcategories of the category of all (say, left) modules over an arbitrary ring whose object classes are closed under direct limit, direct product and pure substructure were called definable by Crawley-Boevey. I will report on some joint results with Ivo Herzog on such subcategories.

365 FIFTH AVE
NEW YORK, NY 10016
philipp.rothmaler at bcc.cuny.edu

Weighted Grothendieck groups

Hans Schoutens, City University of New York

Grothendieck groups and rings may come in many guises: as the $K_0$ -theory of projective modules, as the Grothendieck ring of varieties over a field, as the Euler characteristic in Euclidean topology, etc. In most cases, however, they are insensitive to the finer structure (like torsion, nilpotents, etc.). By introducing a weighted version, I will show that in the module case, we can counteract unwanted cancellation, leading to interesting new invariants of a ring or a scheme.

NEW YORK
hschoutens at citytech.cuny.edu

Finiteness conditions on Leavitt path algebra

Mercedes Siles Molina, Universidad de Malaga

Leavitt path algebras of row-finite graphs have being introduced very recently. They are the algebraic relatives of graph C*-algebras and provide us with examples of rings whose algebraic structure is determined by highly visual properties of the underlying graph. In this talk we will introduce Leavitt path algebras and will give classification theorems for those which satisfy certain finiteness conditions (such as being finite or left artinian).

CAMPUS DE TEATINOS
MALAGA, 29071 MALAGA
SPAIN
msilesm at uma.es

Notes

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