Existence of a point

LEMMA 07.10   Let $A$ be a ring. If $A\neq 0$ (which is equivalent to saying that $1_A\neq 0_A$), then we have $\operatorname{Spec}(A)\neq \emptyset$.

PROOF.. Assume $A\neq 0$. Then by Zorn's lemma we always have a maximal ideal $\mathfrak{m}$ of $A$. A maximal ideal is a prime ideal of $A$ and is therefore an element of $\operatorname{Spec}(A)$. $\qedsymbol$

LEMMA 07.11   Let $A$ be a ring, $f$ be its element. We have $O_f=\emptyset$ if and only if $f$ is nilpotent.

PROOF.. We have already seen that $A_f=0$ if and only if $f$ is nilpotent. (Corollary 7.9). Since $O_f$ is homeomorphic to $\operatorname{Spec}(A_f)$, we have the desired result. $\qedsymbol$