Algebras having bases that consist solely of strongly regular elements

Abstract

“Locally invertible” algebras, those algebras which have a basis consisting solely of strongly regular elements, are introduced as a generalization of “invertible algebras,” that is, algebras which have a basis consisting solely of units. While this new family properly contains the family of (necessarily unital) invertible algebras, its definition does not assume the existence of a multiplicative identity. Because of this, we consider both unital and non-unital examples of locally invertible algebras. In particular, we show that under a mild condition on the basis of a not necessarily unital R-algebra A, the R-algebras Mn(A) of finite matrix rings over the R-algebra A. Furthermore, many infinite matrix algebras are also locally invertible, but not all. Also it is shown that all semiperfect D-algebras over a division ring D are locally invertible.