Ring Theory Seminar - Cosmin Roman

November 6, 2019
Friday, November 8, 2019 - 4:45pm to 5:45pm
Cockins Hall 240
Ring Theory Seminar

Title: Nonsingular and Cononsingular Properties via Endomorphism Ring of Modules

Speaker: Cosmin Roman -  The Ohio State University, Lima

Abstract: An important result in Ring Theory, relating two important notions: extending rings and Baer rings, is the Chatters-Khuri theorem (1980) stating that "A ring is extending and nonsingular if and only if the ring is Baer and cononsingular". The two auxiliary properties that are invoked are very useful, and warranted study in various other settings (nonsingular rings are a staple notion in many research papers). More recently, with the advent of Baer property of modules, and the extension of Chatters-Khuri result to a module-theoretic setting, both nonsingular and cononsingular properties have been re-defined for modules, via the endomorphism ring of said modules. Modules with the new properties are dubbed $\mathcal K$-nonsingular module and $\mathcal K$-cononsingular module. While the former property is found to be satisfied quite often in research papers in the context of Baer modules, Rickart modules, and their generalizations, the latter property has not yet been given the same treatment. However, since the concept of $\mathcal K$-cononsingular module generalizes that of extending module, it is worth attempting to correct this oversight. In this presentation, we will study this class of modules, investigating basic properties and presenting related notions. Examples illustrating these notions and delimiting our results will also be provided.  (This is a joint work with F. A. Ebrahim and S. T. Rizvi)

 

 

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