Weierstrass ring

In mathematics, a Weierstrass ring, named by Nagata (1962, section 45) after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal is a finite extension of a regular local ring.

Examples

• The Weierstrass preparation theorem can be used to show that the ring of convergent power series over the complex numbers in a finite number of variables is a Wierestrass ring. The same is true if the complex numbers are replaced by a perfect field with a valuation.
• Every ring that is a finitely-generated module over a Weierstrass ring is also a Weierstrass ring.

References

• Danilov, V. I. (2001) [1994], "W/w097500", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• M. Nagata, "Local rings", Interscience (1962)