Weierstrass ring

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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.

[edit] 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 finite module over a Weierstrass ring is also a Weierstrass ring.

[edit] References

• Danilov, V. I. (2001), “Weierstrass ring”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• M. Nagata, "Local rings" , Interscience (1962)