Support of a module
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
$M$ 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=33863
Support of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_module&oldid=33863
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