Support of a module

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$M$ 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 $M=0$.


[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
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Support of a module. Encyclopedia of Mathematics. URL:
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