Difference between revisions of "Primary ideal"
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''of a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744601.png" />'' | ''of a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744601.png" />'' | ||
| − | An [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744602.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744604.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744605.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744606.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744607.png" />. In the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744608.png" /> of integers a primary ideal is an ideal of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446010.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446011.png" /> is a natural number. In commutative algebra an important role is played by the representation of an arbitrary ideal of a commutative [[Noetherian ring|Noetherian ring]] as an intersection of a finite number of primary ideals — a [[Primary decomposition|primary decomposition]]. More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446012.png" /> denote the set of prime ideals of a [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446013.png" /> that are annihilators of non-zero submodules of a [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446014.png" />. A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446015.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446016.png" /> over a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446017.png" /> is called primary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446018.png" /> is a one-element set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446019.png" /> is commutative, then every proper submodule of a Noetherian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446020.png" />-module that cannot be represented as an intersection of submodules strictly containing it is primary. In the non-commutative case this is not true and therefore attempts have been undertaken to construct various non-commutative generalizations of the notion of primarity. E.g., a proper submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446021.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446022.png" /> is called primary if for every non-zero injective submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446023.png" /> of the [[injective hull]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446024.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446025.png" /> (cf. [[Injective module|Injective module]]) the intersection of the kernels of the homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446027.png" /> is trivial. Another successful generalization is the notion of a tertiary ideal [[#References|[4]]]: A left ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446028.png" /> of a left Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446029.png" /> is called tertiary if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446031.png" />, it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446032.png" /> that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446033.png" />, there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446035.png" />. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446036.png" /> is primary if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446037.png" /> satisfies the Artin–Rees condition: For arbitrary left ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446039.png" /> there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446041.png" /> (cf. [[#References|[3]]]). | + | An [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744602.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744604.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744605.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744606.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744607.png" />. In the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744608.png" /> of integers a primary ideal is an ideal of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446010.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446011.png" /> is a natural number. In commutative algebra an important role is played by the representation of an arbitrary ideal of a commutative [[Noetherian ring|Noetherian ring]] as an intersection of a finite number of primary ideals — a [[Primary decomposition|primary decomposition]]. More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446012.png" /> denote the set of prime ideals of a [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446013.png" /> that are annihilators of non-zero submodules of a [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446014.png" />. A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446015.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446016.png" /> over a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446017.png" /> is called primary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446018.png" /> is a one-element set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446019.png" /> is commutative, then every proper submodule of a Noetherian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446020.png" />-module that cannot be represented as an intersection of submodules strictly containing it is primary. In the non-commutative case this is not true and therefore attempts have been undertaken to construct various non-commutative generalizations of the notion of primarity. E.g., a proper submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446021.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446022.png" /> is called primary if for every non-zero injective submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446023.png" /> of the [[injective hull]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446024.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446025.png" /> (cf. [[Injective module|Injective module]]) the intersection of the kernels of the homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446027.png" /> is trivial. Another successful generalization is the notion of a [[tertiary ideal]] [[#References|[4]]]: A left ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446028.png" /> of a left Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446029.png" /> is called tertiary if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446031.png" />, it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446032.png" /> that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446033.png" />, there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446035.png" />. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446036.png" /> is primary if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446037.png" /> satisfies the Artin–Rees condition: For arbitrary left ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446039.png" /> there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446041.png" /> (cf. [[#References|[3]]]). |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O. Goldman, "Rings and modules of quotients" ''J. of Algebra'' , '''13''' : 1 (1969) pp. 10–47</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O. Goldman, "Rings and modules of quotients" ''J. of Algebra'' , '''13''' : 1 (1969) pp. 10–47</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963)</TD></TR></table> | ||
Revision as of 19:20, 5 October 2017
of a commutative ring 
An ideal 
such that if 
and 
, then either 
or 
for some natural number 
. In the ring 
of integers a primary ideal is an ideal of the form 
, where 
is a prime number and 
is a natural number. In commutative algebra an important role is played by the representation of an arbitrary ideal of a commutative Noetherian ring as an intersection of a finite number of primary ideals — a primary decomposition. More generally, let 
denote the set of prime ideals of a ring 
that are annihilators of non-zero submodules of a module 
. A submodule 
of a module 
over a Noetherian ring 
is called primary if 
is a one-element set. If 
is commutative, then every proper submodule of a Noetherian 
-module that cannot be represented as an intersection of submodules strictly containing it is primary. In the non-commutative case this is not true and therefore attempts have been undertaken to construct various non-commutative generalizations of the notion of primarity. E.g., a proper submodule 
of a module 
is called primary if for every non-zero injective submodule 
of the injective hull 
of the module 
(cf. Injective module) the intersection of the kernels of the homomorphisms from 
into 
is trivial. Another successful generalization is the notion of a tertiary ideal [4]: A left ideal 
of a left Noetherian ring 
is called tertiary if, for any 
, 
, it follows from 
that, for any 
, there is an element 
such that 
. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring 
is primary if and only if 
satisfies the Artin–Rees condition: For arbitrary left ideals 
of 
there is a natural number 
such that 
(cf. [3]).
References
| [1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
| [2] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |
| [3] | O. Goldman, "Rings and modules of quotients" J. of Algebra , 13 : 1 (1969) pp. 10–47 |
| [4] | L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963) |
How to Cite This Entry:
Primary ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_ideal&oldid=39565
Primary ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_ideal&oldid=39565
This article was adapted from an original article by V.T. Markov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article