lie over

lie over (third-person singular simple present lies over, present participle lying over, simple past lay over, past participle lain over)

1. (intransitive) To be deferred to a future occasion.
2. (transitive, algebra, ring theory, of a prime ideal ${\displaystyle q}$ of a ring ${\displaystyle R'}$ with subring ${\displaystyle R}$) To intersect with ${\displaystyle R}$ at some other prime ideal ${\displaystyle p}$; ${\displaystyle q}$ lies over ${\displaystyle p}$ if and only if ${\displaystyle q\cap R=p}$.

   Synonym: lie above

   • 1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Academic Press, page 84:

     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″.

   • 2012, A. Barlotti, E.W. Ellers, P. Plaumann, K. Strambach, Generators and Relations in Groups and Geometries, Springer Science & Business Media, page 369:

     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′.

• overlie, relievo