Regular system of parameters

Definition with symbols
Let $$R$$ be a regular local ring, viz a local ring in which the unique maximal ideal $$\mathcal{m}$$. Let $$d$$ be the Krull dimension of $$R$$. A regular system of parameters for $$R$$ is a collection of elements $$a_1, a_2, \ldots, a_d \in \mathcal{m}$$ satisfying the following equivalent conditions:


 * $$a_1, a_2, \ldots, a_d$$ generate $$\mathcal{m}$$ as an ideal
 * The images of $$a_1, a_2, \ldots, a_d$$ form a basis for $$\mathcal{m}$$ as a vector space

Equivalence of definitions
The equivalence of definitions follows from Nakayama's lemma.