Kommutativer Ring/Ideal/Superhöhe/en/Definition

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

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