The Ohio University-Ohio State University ring theory seminar series hosts two speakers on Friday, March 16.
Louis Rowen (Bar Ilan University, Ramat Gan, Israel) discusses “Exterior Semi-algebras” from 5-5:40 p.m. in Cockins Hall 240, OSU-Columbus.
Abstract: (Joint work with Letterio Gatto) In this talk, we review negation maps and “systems,” and their application to linear algebra in a rather general framework that includes both tropical algebra and hyperfields. The usual definition of Grassmann (exterior) algebras generalizes directly to semi-algebras, and has a built-in negation function for elements of degree \geq 2, so the theory of systems can be applied directly to Gatto’s theory, unifying results of linear algebra from different perspectives including the classical perspective.
Surjeet Singh (Panjab University, Chandigarh, India) discusses “Rings with cyclic modules almost self-injective” from 5:45-6:25 p.m. in Cockins Hall 240, OSU-Columbus.
Abstract: If a ring $R$ is such that every cyclic right $R$-module is injective, as proved by Osofsky in 1964, $R$ is semi-simple artinian. This has motivated others to find structure of rings $R$ over which certain class of modules have a well defined property $P$. For instance, Koehler and Ahsan independently studied rings over which cyclic right modules are quasi-injective, Faith studied rings over which proper cyclic right modules are injective. Baba had introduced the concept of almost relative injectivity in 1989. If a module $M$ is almost $M$-injective, then $M$ is said to be \textit{almost self-injective}. Any quasi-injective module is almost self-injective.
A ring $R$ over which all cyclic right modules are almost self-injective, is called a \textit{right} $cai$-ring. It is proved that
a right noetherian ring $R$ that is a right $cai$, is a finite direct sum of local uniserial rings, serial ring with
the square of its radical zero, and $2\times 2$ matrix ring
$\left[
\begin{array}{cc}
D & M \\
0 & S%
\end{array}%
\right] $, where $D$ is a local, noetherian, serial domain, $S$ a serial ring with $J(S)^{2}$ = 0, $M$ a $(D,S)$-bimodule
such that $M_{S}$ is simple, $_{D}M$ is a torsion-free divisible module and $End(M_{S})$ is the
classical quotient ring of $D$, and certain other conditions. (Joint work with S. K. Jain)
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