  By combining well-known techniques from both noncommutative algebra and
computational commutative algebra, we observe that an algorithmic approach can
be applied to the study of irreducible representations of finitely presented
algebras. In slightly more detail: Assume that $n$ is a positive integer, that
$k$ is a computable field, that $\bar{k}$ denotes the algebraic closure of $k$,
and that $M_n(\bar{k})$ denotes the algebra of $n \times n$ matrices with
entries in $\bar{k}$. Let $R$ be a finitely presented $k$-algebra. Calculating
over $k$, the procedure will (a) decide whether an irreducible representation
$R \to M_n(\bar{k})$ exists, and (b) explicitly construct an irreducible
representation $R \to M_n(\bar{k})$ if at least one exists. (For (b), it is
necessary to assume that $k[x]$ is equipped with a factoring algorithm.) An
elementary example is worked through.
