Ganze Erweiterung/Normale Basis/Going down/Fakt

${\displaystyle {}{\mathfrak {p}}_{1}\subset {\mathfrak {p}}_{2}\,}$

in ${\displaystyle {}R}$ und einem Primideal ${\displaystyle {}{\mathfrak {q}}_{2}}$ in ${\displaystyle {}S}$ mit

${\displaystyle {}{\mathfrak {q}}_{2}\cap R={\mathfrak {p}}_{2}\,}$

ein Primideal ${\displaystyle {}{\mathfrak {q}}_{1}}$ mit

${\displaystyle {}{\mathfrak {q}}_{1}\subset {\mathfrak {q}}_{2}\,}$

und

${\displaystyle {}{\mathfrak {q}}_{1}\cap R={\mathfrak {p}}_{1}\,.}$