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).