The lengths of matrix incidence algebras over small finite fields

The paper considers the problem of computing the lengths of matrix incidence algebras over a field whose cardinality is strictly less than the matrix size $n$. For $n=3,4$, the lengths of all such algebras are determined over the field of two elements. In the case where the ground field and the number $n$ are arbitrary but the Jacobson radical of the algebra has nilpotency index $2$, an upper bound for the length is provided. In addition, the incidence algebras isomorphic to a direct sum of triangular matrix algebras of order $2$ and an algebra of diagonal matrices are considered. It is shown that the lengths of these algebras over the field of two elements can equal only two different numbers, which can be determined explicitly. Moreover, the diagonal number of a matrix incidence algebra is introduced and bounded above.