Krull ring

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


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