  The rank of a ring $R$ is the supremum of minimal cardinalities of generating
sets of $I$ as $I$ ranges over ideals of $R$. Matson showed that every positive
integer occurs as the rank of some ring $R$. Motivated by the result of Cohen
and Gilmer that a ring of finite rank has Krull dimension $0$ or $1$, we give
four different constructions of rings of rank $n$ (for all positive integers
n). Two constructions use one-dimensional domains, and the former of these
directly generalizes Matson's construction. Our third construction uses
Artinian rings (dimension zero), and our last construction uses polynomial
rings over local Artinian rings (dimension one, irreducible, not a domain).
