A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about zero-dimensional rings.
I could not find any information about rings of higher Krull dimension. If the problem for general commutative rings is too hard, how about the case when $R$ is Noetherian and/or an integral domain?