Commutative ring/Ideal/Superheight/Definition

Superheight

Let ${}R$ be a noetherian commutative ring and let ${}I\subseteq R$ be an ideal. The (noetherian) superheight is the supremum

$\operatorname {sup} {\left(\operatorname {ht} {\left(IS\right)}:\,S{\text{ is a notherian }}R-{\text{algebra}}\right)}.$