Finding the radical of an algebra of linear transformations

We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle for solving semilinear equations, is polynomial. As an application of the technique we present a simple test for isomorphism of semisimple modules.