Journal article
On invertible algebras
Journal of algebra, Vol.538, pp.1-34
11/15/2019
DOI: 10.1016/j.jalgebra.2019.07.005
Abstract
An algebra A over a field K is said to be invertible if it has a basis B consisting only of units; if B−1 is again a basis, A is invertible-2, or I2. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by López-Permouth, Moore, Szabo, Pilewski [13], [14]. In this paper, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field K(x) which strictly contains K[x], with K an algebraically closed field, has a symmetric basis B=B−1. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields K≠F2 satisfying some additional mild condition. We also show that every commutative affine invertible algebra is almost I2 in the sense that it becomes I2 after localization at a single element.
Details
- Title: Subtitle
- On invertible algebras
- Creators
- Jeremy Edison - University of IowaMiodrag C Iovanov - Romanian Academy
- Resource Type
- Journal article
- Publication Details
- Journal of algebra, Vol.538, pp.1-34
- DOI
- 10.1016/j.jalgebra.2019.07.005
- ISSN
- 0021-8693
- eISSN
- 1090-266X
- Publisher
- Elsevier Inc
- Grant note
- name: CNCS — UEFISCDI, award: PN-III-P1-1.1-TE-2016-0124; name: Romanian Department of Research and Innovation, award: PNCDI III
- Language
- English
- Date published
- 11/15/2019
- Academic Unit
- Mathematics
- Record Identifier
- 9984241148802771
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