The authors present a method for finding the Jacobson radical of an algebra \(A\) of linear transformations acting on a vector space \(W\) of dimension \(n\). They think of such an algebra as a subalgebra of \(M_n(F)\), the algebra of all \(n\times n\) matrices over a field \(F\).
Their method brings the problem back to solving a set of semilinear equations over \(F\). If the characteristic of \(F\) is zero, there is a standard method for doing this. However, if \(F\) has positive characteristic \(p\), this method does not work. Thus the authors show how to obtain a descending series of ideals \((A_i)\) \(i\geq 1\) which after \(\log_p(n)\) steps gives the radical.
In the second section of the paper, the authors describe the ideals \(A_i\) and show some of their properties. In the third section of the paper, they consider representations of \(A\) on an arbitrary finite dimensional vector space and how they decompose into composition factors. They also find invariants of semisimple representations of \(A\). In the final section they sketch how the results obtained might be used in the development of new algorithms for determining the radical of an algebra \(A\) as well as an isomorphism of semisimple \(A\) modules.