The Ohio University-Ohio State University ring theory seminar series presents Jeremy Edison discussing “Boyle’s conjecture and perfect localizations” on Friday, April 19, at 4:45 p.m. in Cockins Hall 240, OSU-Columbus.
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 L\’opez-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 $\mathcal{B}$ consisting entirely of units. If $\mathcal{B}^{-1} = \{ b^{-1} : b \in \mathcal{B} \}$ 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 investigate this question and these algebras in the commutative setting. In particular, we show that a version of the classical Noether Normalization Lemma holds for commutative, finitely generated invertible algebras. We use this to prove that if $A$ is a commutative, finitely generated, invertible algebra with no zero divisors, then $A$ is “almost I2,” in the sense that one may obtain an I2 algebra from $A$ after localization at a single element. We also investigate invertibility and the I2 property for commutative algebras of Krull dimension 1.
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