Difference between revisions of "Weierstrass ring"
From Encyclopedia of Mathematics
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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. | 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. | ||
====References==== | ====References==== | ||
<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> | <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> | ||
Revision as of 15:26, 1 May 2014
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, cf. Analytic ring) over a perfect field, to which the Weierstrass preparation theorem (cf. Weierstrass 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=32070
Weierstrass ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_ring&oldid=32070
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article