On Frobenius full matrix algebras with structure systems

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.