# Meaning of sentence in EGA I, Section 1.7.

In Section 1.7 of EGA I, we have the following:

(1.7.1) Étant donné un $$A$$-module $$M$$, on appelle support de $$M$$ et on note $$\text{Supp}(M)$$ l'ensemble des idéaux premiers $$\mathfrak{p}$$ de $$A$$ tels que $$M_{\mathfrak{p}} \ne 0$$. Pour que $$M = 0$$, il faut et il suffit que $$\text{Supp}(M) = \emptyset$$, car si $$M_{\mathfrak{p}}= 0$$ pour tout $$\mathfrak{p}$$, l'annulateur d'un élément $$x \in M$$ ne peut être contenu dans aucun idéal premier de $$A$$, donc est $$A$$ tout entier.

I would translate this as follows:

Being given an $$A$$-module $$M$$, we say the {\it support} of $$M$$ and we write $$\text{Supp}(M)$$ for the set of prime ideals $$\mathfrak{p}$$ of $$A$$ such that $$M_{\mathfrak{p}} \ne 0$$. For $$M = 0$$, it is necessary and sufficient that $$\text{Supp}(M) = \emptyset$$, because if $$M_{\mathfrak{p}} = 0$$ for every $$\mathfrak{p}$$, the annihilator of an element $$x \in M$$ cannot be contained in any prime ideal of $$A$$, therefore $$A$$ is [???].

As you can see, I am stuck on the phrase "tout entier". It sounds like "completely integral", but what does this actually mean in this context?

• "...therefore is equal to $A$." Feb 13, 2018 at 20:45
• "donc l'annulateur est A tout entier" Feb 13, 2018 at 20:47

Thanks to the comments by user26857 and Billy, I see now that it's just saying the annihilator must be all of $A$.