Difference between revisions of "Support of a module"
From Encyclopedia of Mathematics
(Importing text file) |
(Category:Associative rings and algebras) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | '' | + | ''$M$ over a commutative ring $R$'' |
| − | The set of all prime ideals | + | The set of all prime ideals $\mathfrak{p}$ of $A$ for which the localizations $M_{\mathfrak{p}}$ of the module are non-zero (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]). This set is denoted by $\mathrm{Supp}(M)$. It is a subset of the spectrum of the ring (cf. [[Spectrum of a ring|Spectrum of a ring]]). For example, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime ideals $(p)$, where $p$ divides the order of $M$. For an arbitrary module $M$ the set $\mathrm{Supp}(M)$ is empty if and only if $M=0$. |
====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></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> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
| + | |||
| + | [[Category:Associative rings and algebras]] | ||
Latest revision as of 20:46, 18 October 2014
$M$ over a commutative ring $R$
The set of all prime ideals $\mathfrak{p}$ of $A$ for which the localizations $M_{\mathfrak{p}}$ of the module are non-zero (cf. Localization in a commutative algebra). This set is denoted by $\mathrm{Supp}(M)$. It is a subset of the spectrum of the ring (cf. Spectrum of a ring). For example, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime ideals $(p)$, where $p$ divides the order of $M$. For an arbitrary module $M$ the set $\mathrm{Supp}(M)$ is empty if and only if $M=0$.
References
| [1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
How to Cite This Entry:
Support of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_module&oldid=16164
Support of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_module&oldid=16164
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article