Algebras having bases that consist solely of units
The authors study associative algebras \(A\) that contain a subalgebra \(R\) such that \(A\) is a free left \(R\)-module (they call these notions \(R\)-algebras) and there exists a basis \(B\) of \(A\) with the property that all its elements are invertible (an invertible basis). This paper extends and completes the result given in [the authors, Contemp. Math. 499, 219--228 (2009; Zbl 1191.16037)]. The authors classify this type of algebras with some additional properties over the invertible basis \(B\). For example they prove that an \(R\)-algebra \(A\) is a crossed product if and only if \(A\) contains an invertible basis \(B\) such that for every \(v\in B\) there exists \(\alpha\in U(R)\) with \(\alpha v^{-1}\in B\) and for every \(v\in B\) there exists \(\sigma_v\in \mathrm{Aut}(R)\) such that \(vr=\sigma_v(r)v\) for every \(r\in R\). Moreover, they prove that if \(A\) is an algebra over a ring \(R\) with a basis that includes \(1\), then \(M_n(A)\) is an (\(I2\)) algebra over \(R\), for \(n \geq 2\). Finally, they study finite-dimensional algebras over division rings and some invertible infinite-dimensional algebras.
scientific article; zbMATH DE number 6509138
scientific article