# Lying-Over Theorem

Let $\mathfrak{o}$ be a subring of a commutative ring $\mathfrak{O}$ with nonzero unity and integral over $\mathfrak{o}$.  If $\mathfrak{a}$ is an ideal of $\mathfrak{o}$ and $\mathfrak{A}$ an ideal of $\mathfrak{O}$ such that

 $\mathfrak{A\cap o\;=\;a},$

then $\mathfrak{A}$ is said to lie over $\mathfrak{a}$.

Theorem.  If $\mathfrak{p}$ is a prime ideal of a ring $\mathfrak{o}$ which is a subring of a commutative ring $\mathfrak{O}$ with nonzero unity and integral over $\mathfrak{o}$, then there exists a prime ideal $\mathfrak{P}$ of $\mathfrak{O}$ lying over $\mathfrak{p}$.  If the prime ideals $\mathfrak{P}$ and $\mathfrak{Q}$ both lie over $\mathfrak{p}$ and  $\mathfrak{P\,\subseteq\,Q}$,  then  $\mathfrak{P\,=\,Q}$.

## References

• 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals.  Academic Press, New York (1971).
• 2 P. Jaffard: Les systèmes d’idéaux.  Dunod, Paris (1960).
Title Lying-Over Theorem LyingOverTheorem 2013-03-22 19:15:42 2013-03-22 19:15:42 pahio (2872) pahio (2872) 6 pahio (2872) Theorem msc 16D99 msc 13C99 lie over