2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64300Let 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).AMS-TeX; 13 pages; no figures. Revised version. To appear in Journal of Pure and Applied AlgebraRings and AlgebrasAlgebraic Geometry16Z05; 14Q20text