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 20:30, 30 October 2016
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) |
Primary ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_ideal&oldid=11634