  Let n be a positive integer, and let R be a finitely presented (but not
necessarily finite dimensional) associative algebra over a computable field. We
examine algorithmic tests for deciding (1) if every n-dimensional
representation of R is semisimple, and (2) if there exist nonsplit extensions
of non-isomorphic irreducible R-modules whose dimensions sum to no greater than
n. Our basic strategy is to reduce each of the considered representation
theoretic decision problems to the problem of deciding whether a particular set
of commutative polynomials has a common zero. Standard methods of computational
algebraic geometry can then be applied (in principle).
