lie over

Definitions

1. 1

   To be deferred to a future occasion.

2. 2

   Of a prime ideal q of a ring R' with subring R, to intersect with R at some other prime…

   Of a prime ideal q of a ring R' with subring R, to intersect with R at some other prime ideal p; q lies over p if and only if q∩R=p.

   • Let R be a subring of a ring R′ which is integral over R. If P is a prime ideal of R, then there is a prime ideal P′ of R′ that lies over P. Moreover, if P′ and P″ are prime ideals of R′ that lie over P and if P′ ⊆ P″, then P′ = P″.
   • Let R ⊆ S be integral domains such that R is integrally closed and S is integral over R. Suppose that P′ ⊆ P are prime ideals of R. Let Q be a prime ideal of S which lies over P. Then Q contains a prime ideal Q′ of S that lies over P′.

The neighborhood