lie over

From Wiktionary, the free dictionary
Jump to navigation Jump to search

English

[edit]

Verb

[edit]

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 of a ring with subring ) To intersect with at some other prime ideal ; lies over if and only if .
    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′.

Anagrams

[edit]