Difference between revisions of "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 | + | {{TEX|done}} |
| + | 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=32341
Lasker ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lasker_ring&oldid=32341
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article