Title: Commutative Algebras Having a Basis of Units
Speaker: Jeremy Edison (University of Iowa)
Abstract: Following López-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.
