(related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
By assumption, \(\varphi \colon R\longrightarrow S\,\) is a ring homomorphism between two commutative rings and \[\varphi ^{ * }\colon \cases{\operatorname {Spek} \left(S\right)\longrightarrow \operatorname {Spek} \left(R\right),\cr J\longmapsto \varphi ^{ * }(J)}\] is the corresponding spectrum function. For the fiber of a prime ideal \(I\in \operatorname {Spek} \left(R\right)\) under the spectrum function, we want to show the following properties:
This follows immediately from the proposition.
For a prime ideal \({J}\lhd S\) we have \(\varphi ^{-1}({J})={I}\) if and only if
The first condition is equivalent to \({I}S\lhd {J}\) und the second to \(\varphi (R\setminus {I})\cap {J}=\emptyset \,\).