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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 $R$'' |
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− | 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]] $I$ of $R$ such that if $a,b \in R$ and $ab \in I$, then either $b \in I$ or $a^n \in I$ for some natural number $n$. In the ring $\mathbf{Z}$ of integers a primary ideal is an ideal of the form $p^n\mathbf{Z}$, where $p$ is a prime number and $n$ 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 $\mathrm{Ass}(M)$ denote the set of prime ideals of a [[ring]] $R$ that are annihilators of non-zero submodules of a [[module]] $M$. A submodule $N$ of a module $M$ over a Noetherian ring $R$ is called primary if $\mathrm{Ass}(M/N)$ is a one-element set. If $R$ is commutative, then every proper submodule of a Noetherian $R$-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 $N$ of a module $M$ is called primary if for every non-zero injective submodule $E_1$ of the [[injective hull]] $E$ of the module $M/N$ (cf. [[Injective module]]) the intersection of the kernels of the homomorphisms from $E$ into $E_1$ is trivial. Another successful generalization is the notion of a [[tertiary ideal]] [[#References|[4]]]: A left ideal $I$ of a left Noetherian ring $R$ is called tertiary if, for any $a\in R$, $b \in R\setminus I$, it follows from $aRb \subseteq I$ that, for any $c \in R/I$, there is an element $d \in Rc \setminus I$ such that $aRd \subseteq I$. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring $R$ is primary if and only if $R$ satisfies the Artin–Rees condition: For arbitrary left ideals $I,J$ of $R$ there is a natural number $n$ such that $I^n \cap J \subseteq IJ$ (cf. [[#References|[3]]]). |
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| ====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> |
| + | |
| + | {{TEX|done}} |
Latest revision as of 19:51, 5 October 2017
of a commutative ring $R$
An ideal $I$ of $R$ such that if $a,b \in R$ and $ab \in I$, then either $b \in I$ or $a^n \in I$ for some natural number $n$. In the ring $\mathbf{Z}$ of integers a primary ideal is an ideal of the form $p^n\mathbf{Z}$, where $p$ is a prime number and $n$ 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 $\mathrm{Ass}(M)$ denote the set of prime ideals of a ring $R$ that are annihilators of non-zero submodules of a module $M$. A submodule $N$ of a module $M$ over a Noetherian ring $R$ is called primary if $\mathrm{Ass}(M/N)$ is a one-element set. If $R$ is commutative, then every proper submodule of a Noetherian $R$-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 $N$ of a module $M$ is called primary if for every non-zero injective submodule $E_1$ of the injective hull $E$ of the module $M/N$ (cf. Injective module) the intersection of the kernels of the homomorphisms from $E$ into $E_1$ is trivial. Another successful generalization is the notion of a tertiary ideal [4]: A left ideal $I$ of a left Noetherian ring $R$ is called tertiary if, for any $a\in R$, $b \in R\setminus I$, it follows from $aRb \subseteq I$ that, for any $c \in R/I$, there is an element $d \in Rc \setminus I$ such that $aRd \subseteq I$. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring $R$ is primary if and only if $R$ satisfies the Artin–Rees condition: For arbitrary left ideals $I,J$ of $R$ there is a natural number $n$ such that $I^n \cap J \subseteq IJ$ (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=42013
This article was adapted from an original article by V.T. Markov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article