The American Mathematical Society Chapter at Ohio University presents a Graduate Student Research Seminar featuring Jeremy Edison discussing “Invertible Rings and Algebras” on Thursday, April 18, at 4:30 p.m. in Morton 326.
Jeremy Edison
Edison is currently a graduate student at the University of Iowa, studying under Dr. Miodrag Iovanov, and is planning on graduating in Spring 2019. He earned his M.S. from Iowa in 2016 and B.A. in mathematics from Knox College in 2014. His research interests are in algebra and representation theory, specifically ring theory, linear algebra, Hopf algebras, and he also maintains an interest in mathematics education.
Abstract: Following Lopez-Permouth, Moore, and Szabo (2009) and those authors together with Pilewski (2015), we call an algebra A over a field K invertible if A has a basis β consisting entirely of units. If β-1 = {b-1 : b ε β} is again a basis, we say A is an invertible-2, or I2 algebra. The question of whether an invertible algebra is necessarily I2 arises naturally. We provide several positive results in this direction, and provide various conditions sufficient for detecting invertibility and the I2 property. In particular, given any field extension L/K, L will be I2 as a K-algebra, and if K is algebraically closed then any subalgebra A of the field of rational functions that properly contains the polynomials has an I2 basis. We then use these results to show that semiprimary algebras satisfying a few additional mild conditions also have the I2 property. This is joint work with Miodrag Iovanov.
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