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Difference between revisions of "Weierstrass ring"

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A local Hensel pseudo-geometric ring (cf. [[Geometric ring|Geometric ring]]; [[Hensel ring|Hensel ring]]) each quotient ring of which by a prime ideal is a finite extension of a regular [[Local ring|local ring]] (cf. also [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]). A Weierstrass ring is analytically irreducible. Any finite extension of a Weierstrass ring is a Weierstrass ring. Examples of Weierstrass rings are analytic rings (rings of convergent power series, cf. [[Analytic ring|Analytic ring]]) over a perfect field, to which the Weierstrass preparation theorem (cf. [[Weierstrass theorem|Weierstrass theorem]]) is applicable.
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A local Hensel pseudo-geometric ring (cf. [[Geometric ring]]; [[Hensel ring]]) each quotient ring of which by a prime ideal is a finite extension of a regular [[local ring]] (cf. also [[Regular ring (in commutative algebra)]]). A Weierstrass ring is analytically irreducible. Any finite extension of a Weierstrass ring is a Weierstrass ring. Examples of Weierstrass rings are [[analytic ring]]s (rings of convergent power series) over a perfect field, to which the [[Weierstrass preparation theorem]] is applicable.
  
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Seydi,  "Sur la théorie des anneaux de Weierstrass I"  ''Bull. Soc. Math. France'' , '''95'''  (1971)  pp. 227–235</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  H. Seydi,  "Sur la théorie des anneaux de Weierstrass I"  ''Bull. Soc. Math. France'' , '''95'''  (1971)  pp. 227–235</TD></TR>
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Latest revision as of 17:52, 10 January 2016

A local Hensel pseudo-geometric ring (cf. Geometric ring; Hensel ring) each quotient ring of which by a prime ideal is a finite extension of a regular local ring (cf. also Regular ring (in commutative algebra)). A Weierstrass ring is analytically irreducible. Any finite extension of a Weierstrass ring is a Weierstrass ring. Examples of Weierstrass rings are analytic rings (rings of convergent power series) over a perfect field, to which the Weierstrass preparation theorem is applicable.

References

[1] M. Nagata, "Local rings" , Interscience (1962)
[2] H. Seydi, "Sur la théorie des anneaux de Weierstrass I" Bull. Soc. Math. France , 95 (1971) pp. 227–235
How to Cite This Entry:
Weierstrass ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_ring&oldid=15869
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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