In several places in EGA I (for example, statement (1.5.8), which I reproduce below), it says that a ring is "réduit à 0", which I would translate "reduced at 0", but which might also be translated "reduced to 0"? What does this mean? Is it just saying that the nilradical is 0?
Here is an example passage using this phrase:
"Soit $A$ un sous-anneau d'un anneau $B$; pour tout idéal premier minimal $\mathfrak{p}$ de $A$, il existe un idéal premier minimal $\mathfrak{q}$ de $B$ tel que $\mathfrak{p} = A \cap \mathfrak{q}$. En effet, $A_{\mathfrak{p}}$ est un sous-anneau de $B_{\mathfrak{p}}$ (1.3.2) et possède un seul idéal premier $\mathfrak{p}'$ (1.2.6); comme $B_{\mathfrak{p}}$ n'est pas réduit à 0, il possède au moins un idéal premier $\mathfrak{q}'$ et on a nécessairement $\mathfrak{q}' \cap A_{\mathfrak{p}} = \mathfrak{p}'$; l'idéal premier $\mathfrak{q}_1$ de $B$, image réciproque de $\mathfrak{q}'$ est donc tel que $\mathfrak{q}_1 \cap A = \mathfrak{p}$, et a fortiori on a $\mathfrak{q} \cap A = \mathfrak{p}$ pour tout idéal premier minimal $\mathfrak{q}$ de $B$ contenu dans $\mathfrak{q}_1$."
I hope this provides the necessary context.
It just means "is the zero ring." For instance in your quoted passage, every nonzero ring has at least one maximal (hence prime) ideal by Zorn's Lemma. But the zero ring has no proper ideals, hence no prime ideals.