I will discuss new techniques for computing with matrix algebras and their modules over arbitrary fields. The key is a deterministic algorithm to construct a non-nilpotent element in any weakly-closed subset of a non-nilpotent matrix algebra. There are immediate applications of this construction to the problem of testing for isomorphism between two modules, and to obtaining certain decompositions of a module. The methods are quite elementary, requiring almost no structural knowledge of the algebra. Thus the hope is that the resulting algorithms are practicable for a broad range of fields, and I am particularly keen to discuss their practical potential during my visit to Sydney. (This is joint work with Gene Luks at the University of Oregon.)