# Support of a module

From Encyclopedia of Mathematics

*$M$ 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

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