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Ring Theory Seminar - César Alejandro Arellano

Ring Theory Seminar
November 22, 2019
4:45PM - 5:45PM
Cockins Hall 240

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Add to Calendar 2019-11-22 16:45:00 2019-11-22 17:45:00 Ring Theory Seminar - César Alejandro Arellano Title: Non-isomorphic basic modules and domains of divisibility  Speaker: César Alejandro Arellano - UNAM, Mexico and Ohio University Center of Ring Theory and its Applications Abstract: This talk touches upon a couple of topics of recurring interest at the Ohio University Center of Ring Theory and its Applications: they are amenable bases of infinite dimensional algebras, and systems to gauge the injectivity of modules over arbitrary rings.   In the context of amenable bases of infinite dimensional algebras and the modules they induce (the so-called basic modules), we answer (in the negative) the question of whether all basic modules must be isomorphic. A sufficient condition for basic modules to be isomorphic (mutual congeniality of the respective bases) has been the subject of much attention but the question of their relation without further hypotheses had laid dormant until now. The problem in tackling this problem arises because in order to guarantee that two modules are not isomorphic one must identify a module-theoretic property of one not shared by the other; very little is known about any properties that basic modules may or not have.It is with the above purpose that we introduce the divisibility profile of a ring, assigning to each module its divisibility domain, and to the ring the set of all divisibility domains of its modules.   The process of divisibility gauging resembles the various ways to measure homological properties introduced in the past by members and visitors of the CRA and ring theorists elsewhere. In fact, injectivity alone can be measured via injectivity domains, injectivity subdomains, weak-injectivity domains, etc. The introduction of yet another profile that gauges the extent of injectivity of modules in our study is justified by the fact that the basic modules we manage to differentiate in our project actually share the same injectivity domain; it is indeed necessary to use an alternative approach to set them apart.   Our project has allowed us to take a first look at this new profile and its properties. In particular, we will report on an intriguing lack of alignment between the profiles of injectivity and divisibility of the algebra F[x] of polynomials over a field F. In fact, we show that there is a tremendous diversity among the divisibility domains which modules that share the same injectivity domain can have.    This presentation is a report of ongoing research with Sergio R. López-Permouth. Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public

Title: Non-isomorphic basic modules and domains of divisibility 

Speaker: César Alejandro Arellano - UNAM, Mexico and Ohio University Center of Ring Theory and its Applications

Abstract: This talk touches upon a couple of topics of recurring interest at the Ohio University Center of Ring Theory and its Applications: they are amenable bases of infinite dimensional algebras, and systems to gauge the injectivity of modules over arbitrary rings.
 
In the context of amenable bases of infinite dimensional algebras and the modules they induce (the so-called basic modules), we answer (in the negative) the question of whether all basic modules must be isomorphic. A sufficient condition for basic modules to be isomorphic (mutual congeniality of the respective bases) has been the subject of much attention but the question of their relation without further hypotheses had laid dormant until now. The problem in tackling this problem arises because in order to guarantee that two modules are not isomorphic one must identify a module-theoretic property of one not shared by the other; very little is known about any properties that basic modules may or not have.It is with the above purpose that we introduce the divisibility profile of a ring, assigning to each module its divisibility domain, and to the ring the set of all divisibility domains of its modules.
 
The process of divisibility gauging resembles the various ways to measure homological properties introduced in the past by members and visitors of the CRA and ring theorists elsewhere. In fact, injectivity alone can be measured via injectivity domains, injectivity subdomains, weak-injectivity domains, etc. The introduction of yet another profile that gauges the extent of injectivity of modules in our study is justified by the fact that the basic modules we manage to differentiate in our project actually share the same injectivity domain; it is indeed necessary to use an alternative approach to set them apart.
 
Our project has allowed us to take a first look at this new profile and its properties. In particular, we will report on an intriguing lack of alignment between the profiles of injectivity and divisibility of the algebra F[x] of polynomials over a field F. In fact, we show that there is a tremendous diversity among the divisibility domains which modules that share the same injectivity domain can have.
 
 This presentation is a report of ongoing research with Sergio R. López-Permouth.

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