Noetherian local ring

A Noetherian local ring is a Noetherian ring with a unique maximal ideal.

An ideal $I$ of a Noetherian local ring $(A,\mathfrak{m})$ is an ideal of definition when $A/I$ has Krull dimension zero. Equivalently, $I$ is $\mathfrak{m}$-primary, so $\sqrt{I}=\mathfrak{m}$ and ${\mathfrak{m}}^r\subseteq I$ for some positive integer $r$.