Title: Algebras having bases that consist entirely of units
Speaker: Sergio Lopez-Permouth, Ohio University, Athens
Seminar Type: OSU-OU Ring Theory
Abstract: We consider algebras that have bases consisting entirely of units, called invertible algebras. Among other results, it is shown that all finite dimensional algebras over a field other than the binary field \(F_2\) have this property. Also, Invertible finite dimensional algebras over F_2 are fully characterized. An earlier result that M_n(R) is an invertible R-algebra over an arbitrary ring R is extended here to show that if A is any R-algebra which is free as an R-module (and has a basis containing the element 1 \in R then M_n(A) is an invertible R-algebra for any n \ge 2. Various families of algebras, including group rings and crossed products, are characterized in terms of invertibility. In addition, invertibility of infinite dimensional algebras is explored and connections to the absence of the Invariant Basis Number (IBN) property are considered. (This talk is based on a paper by L\'opez-Permouth, Moore, Pilewski and Szabo.)
