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
Canonical name	LyingOverTheorem
Date of creation	2013-03-22 19:15:42
Last modified on	2013-03-22 19:15:42
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Theorem
Classification	msc 16D99
Classification	msc 13C99
Defines	lie over