OU-OSU seminar speakers for 2004-2005
October 22, 2004: Ashish Srivastava, Ohio University,
Quotient rings of algebras of functions and operators
ABSTRACT: We will be speaking on the rings of quotients of algebras
of functions and operators as studied in a paper "Quotient rings of algebras
of functions and operators", by S.K. Jain and A.I.Singh (see Mathematische
Zeitschrift, 234, 721-737, 2000). In this paper, various rings of
quotients of rings B of Banach algebra A-valued continuous functions on a
completely regular Haousdorff space X are constructed in terms of
continuous functions defined on open dense subsets of X. This extends
the results proven by N.J. Fine, I Gillman, and J. Lambeck (1965) for the
case where A is the field of real numbers. The pattern is similar and
utilizes as well as generalizes the results proven for algebras of multipliers
of B by C.A. Akemann, G.K. Pederson, and J. Tomiyama (1973). The techniques
combine those from algebra, analysis, and topology.
November 5, 2004: Warrem McGovern, Bowling Green
StateUniversity, Rings of quotients of C(X)
ABSTRACT: Q(X) and q(X) denote the maximal and classical rings of quotients
of C(X) (resp.) It is known that, since C(X) is a semiprime ring, Q(X)
is always von-Neumann regular. We will characterize when the following
occur:
1) q(X) is a von-Neumann regular ring;
2) C(X) is integrally closed in q(X)
We discuss some "smaller" rings of quotients of C(X) induced by points of
X and show that they share similar properties to q(X) and Q(X).
November 19, 2004: Husain Al-Hazmi, Ohio University,
Some results on nonsingular prime rings. ABSTRACT: A ring is called
right (left) max-min CS if every maximal closed right (left) ideal with nonzero
left (right) annihilator and every minimal closed right (left) ideal of R
is a direct summand of R. In this paper we show, among others, that
if R is a prime ring which is not a domain, then R is right nonsingular,
right max-min CS with uniform right ideal if and only if R is left nonsingular,
left max-min CS with uniform left ideal. The above result gives, in
particular, the Huynh, Jain, L\'opez-Permouth theorem for prime rings of
finite uniform dimension [Proc. Amer. Math.Soc. 129, 3153-4157 (2000)]
January 28, 2005: Adel N. Alahmadi, Ohio University,
Some Results on Semiprime CS Group Algebra.
ABSTRACT: Jain, Kanwar, Malik, and Srivastava proved that the group algebra
$K[D_{\infty }]$ is $CS$ if and only if $char(K)\neq 2$. Then Antonio Behn
proved that if $K[G]$ is prime $CS$-ring with $G$ polycyclic-by-finite, then
$G$ is torsion-free or $G\cong D_{\infty }$ and $char(K)\neq 2$. We proved
that if $K[G]$ is semiprime having no domains summands with $G$ polycyclic-by-finite,
then $K[G]$ is hereditary if and only if it is $CS$. Description of such
group algebra, when $K$ is algebraically closed, is also given.
February 4, 2005: Howard M Thompson, University
of Michigan, Ann Arbor, Log Geometry and Log regular local rings.
ABSTRACT: In 1994, Kato (Toric singularities, Amer J Math 116, no 5, 1073-1099)
introduced the notion of a log regular local ring. These rings extend the
notion of a toric singularity to mixed characteristic. They arise naturally
in log geometry, the theory of schemes equipped with a log structure. Kato's
rings are integrally closed. After a brief discussion of the motivation for
log geometry, we will examine a variation on Kato's definition that removes
the normality requirement. In this setting, we will describe a structure
theorem for the completions of log regular local rings. We will also describe
the valuative log space of a log scheme, a space analogous to the Zariski
abstract Riemann surface of a variety, and state a theorem about the coherence
of its structure sheaf.
February 18, 2005: Thomas G. Kucera, University
of Manitoba, Winnipeg, Canada, The structure of the indecomposable injective
modules over the first Weyl algebra.
ABSTRACT: I will start with some motivation from algebra and from logic
for my long-standing interest in problems of this sort. Most of the
talk will be devoted to results of my PhD student Alina Duca on the subject
of the title. Let \( \Aa \) denote the first Weyl algebra (over a field of
characteristic \( 0 \), usually algebraically closed). Every indecomposable
injective over \( \Aa \) is the injective envelope of a simple module, and
the classification of the simples by Bavula (earlier, Block) gives a classification
of the indecomposables and some basic information about their structure.
Considerations motivated originally from logic (first-order definability),
but which can easily be seen algebraically, naturally associate to each indecomposable
a localization of \( \Aa \) (a ring intermediate between it and the
Weyl division algebra \( \mathbb{D}_{1} \)). This localization is a principal
left and right ideal domain. Inspired by a classic treatment of Ore [1932],
a consideration of the arithmetic of such a ring \( R \) on an element-by-element
basis (rather than by an ideal-theoretic analysis) leads to a very nice description
of the internal structure of the indecomposables. There are several different
unique factorization theorems for \( R \); we can associate a similarity
class of irreducible ring elements \( r \) with each indecomposable \( E
\); and then define the \( n \)-th layer of \( E \) to be those elements
of \( E \) which are annihilated by some ring element whose factorization
has no more than \( n \) irreducibles similar to \( r \). We are then
able to find an even stronger localization \( \bar{R} \) for which these
layers exactly correspond to the socle series of \( E \) with respect to
\( \bar{R} \). The work is not complete: there is a detailed analysis of
the structure of the endomorphism rings of the indecomposables underway;
and some general questions about the nature of localizations intermediate
between the first Weyl algebra and the Weyl division algebra. One of the
purposes of giving this talk is to seek further questions of this sort that
might be answered using these results. Probably some of these results
can be applied to generalized Weyl algebras in the sense of Bavula; unfortunately
very little will lift to Weyl algebras of degree greater t.