  The notion of multiplicity of a module first arose as consequence of
Hilbert's work on commutative algebra, relating the dimension of rings with the
degree of certain polynomials. For noncommutative rings, the notion of
multiplicity first appeared in the context of modules for the Weyl algebra in
Bernstein's solution of the problem of analytic continuation posed by I.
Gelfand. The notion was shown to be useful to many more noncommutative rings,
especially enveloping algebras, rings of differential operators, and quantum
groups. In all these cases, the existence of multiplicity is related to the
existence of Hilbert-Samuel polynomials. In this work we give an axiomatic
definition of algebras with a notion of multiplicity, which we call very nice
and modest algebras. We show, in an abstract setting, how the existence of
Hilbert-Samuel polynomials implies the existence of a notion of multiplicity.
We apply our results for the category of min-holonomic modules -- a notion
which coincides with holonomic modules for simple algebras -- and that shares
many similarities with it. In particular, we generalize the usual results in
the literature that are stated for Ore domains, in the more general context of
prime algebras, and we show that rational Cherednik algebras admit a notion of
multiplicity.
