  We prove a general result on presentations of finitely-generated algebras and
apply it to obtain nice presentations for some noncommutative algebras arising
in the matrix bispectral problem. By "nice presentation" we mean a presentation
that has as few as possible defining relations. This in turn, has potential
applications in computer algebra implementations and examples.
  Our results can be divided into three parts. In the first two, we consider
bispectral algebras with the eigenvalue in the physical equation to be
scalar-valued for $2\times 2$ and $3\times 3$ matrix-valued eigenfunctions. In
the third part, we assume the eigenvalue in the physical equation to be
matrix-valued and draw an important connection with spin Calogero-Moser
systems. In all cases, we show that these algebras are finitely presented. As a
byproduct, we answer positively a conjecture of F.~A.~Gr\"{u}nbaum about these
algebras.
