On Frobenius full matrix algebras with structure systems

Abstract

Let \(n\geq 2\) be an integer. In [5] and  [6], an \(n\times n\) \(\mathbb{A}\)-full matrix algebra over a field \(K\) is defined to be the set   \(\mathbb{M}_n(K)\) of all square \(n\times n\) matrices with coefficients in \(K\) equipped with a multiplication   defined by a structure system \(\mathbb{A}\), that is, an \(n\)-tuple of \(n\times n\) matrices with certain properties. In  [5] and  [6],  mainly \(\mathbb{A}\)-full matrix algebras having \((0,1)\)-structure systems are studied, that is, the structure systems     \(\mathbb{A}\) such that all entries are \(0\) or \(1\). In the present  paper we study \(\mathbb{A}\)-full matrix algebras having non \((0,1)\)-structure systems. In particular, we study the Frobenius \(\mathbb{A}\)-full matrix algebras. Several infinite  families of such algebras with nice  properties are constructed in Section 4.