Krull ring

A Krull ring is a particular type of commutative ring studied in commutative algebra and related branches of mathematics and named after the German mathematician Wolfgang Krull.

Formal definition

Let $A$ be an integral domain and let $P$ be the set of all prime ideals of $A$ of height one. Then $A$ is a Krull ring if and only if
# $A\_\{mathfrak\{p$ is a discrete valuation ring for all $mathfrak\{p\}\; in\; P$, and
# every non-zero principal ideal is the intersection of a finite number of primary ideals of height one.

Examples

# Every normal noetherian domain is a Krull ring.
# If $A$ is a Krull ring then so is the polynomial ring $A\; [x]$ and the formal power series ring $Ax$.
# Let $A$ be a noetherian domain with quotient field $K$, and $L$ be a finite algebraic extension of $K$. Then the integral closure of $A$ in $L$ is a Krull ring.

References

* Hideyuki Matsumura, "Commutative Algebra". Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
* Hideyuki Matsumura, "Commutative Ring Theory". Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp. ISBN 0-521-25916-9