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A [[Commutative ring|commutative ring]] in which any [[Ideal|ideal]] has a [[Primary decomposition|primary decomposition]], that is, can be represented as the intersection of finitely-many primary ideals. Similarly, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057600/l0576001.png" />-module is called a Lasker module if any submodule of it has a primary decomposition. Any module of finite type over a Lasker ring is a Lasker module. E. Lasker [[#References|[1]]] proved that there is a primary decomposition in polynomial rings. E. Noether [[#References|[2]]] established that any [[Noetherian ring|Noetherian ring]] is a Lasker ring.
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A [[Commutative ring|commutative ring]] in which any [[Ideal|ideal]] has a [[Primary decomposition|primary decomposition]], that is, can be represented as the intersection of finitely-many primary ideals. Similarly, an $A$-module is called a Lasker module if any submodule of it has a primary decomposition. Any module of finite type over a Lasker ring is a Lasker module. E. Lasker [[#References|[1]]] proved that there is a primary decomposition in polynomial rings. E. Noether [[#References|[2]]] established that any [[Noetherian ring|Noetherian ring]] is a Lasker ring.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lasker,  "Zur Theorie der Moduln und Ideale"  ''Math. Ann.'' , '''60'''  (1905)  pp. 19–116</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Noether,  "Idealtheorie in Ringbereiche"  ''Math. Ann.'' , '''83'''  (1921)  pp. 24–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lasker,  "Zur Theorie der Moduln und Ideale"  ''Math. Ann.'' , '''60'''  (1905)  pp. 19–116</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Noether,  "Idealtheorie in Ringbereiche"  ''Math. Ann.'' , '''83'''  (1921)  pp. 24–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>

Latest revision as of 11:43, 29 June 2014

A commutative ring in which any ideal has a primary decomposition, that is, can be represented as the intersection of finitely-many primary ideals. Similarly, an $A$-module is called a Lasker module if any submodule of it has a primary decomposition. Any module of finite type over a Lasker ring is a Lasker module. E. Lasker [1] proved that there is a primary decomposition in polynomial rings. E. Noether [2] established that any Noetherian ring is a Lasker ring.

References

[1] E. Lasker, "Zur Theorie der Moduln und Ideale" Math. Ann. , 60 (1905) pp. 19–116
[2] E. Noether, "Idealtheorie in Ringbereiche" Math. Ann. , 83 (1921) pp. 24–66
[3] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
How to Cite This Entry:
Lasker ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lasker_ring&oldid=15831
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article