For questions regarding a commutative Noetherian ring whose localization at each prime ideal is a regular local ring.
We say that a commutative Noetherian ring $R$ is regular whenever its localization at every prime ideal $P$ is a regular local ring $(R_P, P R_P, k(P))$ in the sense that the Krull dimension of $R_P$ is equal to the unique number of generators of the maximal ideal $P R_P$ of $R_P,$ i.e., we have that $$\dim R_P = \mu(P R_P) = \dim_{k(P)} \frac{P R_P}{(P R_P)^2},$$ where $\dim R_P$ is Krull dimension, and the second dimension is the $k(P)$-vector space dimension.
Regular rings enjoy a wealth of desirable properties, one of the most fundamental of which is that every localization of a regular ring is a regular local ring. Every field is a regular ring of Krull dimension zero. Further, a polynomial ring $k[x_1, \dots, x_n]$ over a field is a regular ring of Krull dimension $n.$ Dedekind domains are another common example of regular rings.
