Regular local ring

There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity.

Definition

Let $A$ be a Noetherian local ring with maximal ideal $m$ and residual field $k=A/m$. The following conditions are equivalent:

1. The Krull dimension of $A$ is equal to the dimension of $m/m^{2}$ as a $k$-vector space.

And when these conditions hold, $A$ is called a regular local ring.