Algebras having bases that consist solely of units

Algebras having bases that consist entirely of units (called invertible algebras) are studied. Among other results, it is shown that all finite-dimensional algebras over a field other than the binary field F2 have this property. Invertible finite-dimensional algebras over F2 are fully characterized. Examples of invertible algebras are shown to include all (non-trivial) matrix rings over arbitrary algebras. In addition, various families of algebras, including group rings and crossed products, are characterized in terms of invertibility. Invertibility of infinite-dimensional algebras is also explored.