The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.27.11. Let $R$ be a ring.

  1. An ideal maximal among the ideals which do not contain a nonzerodivisor is prime.

  2. If $R$ is nonzero and every nonzero prime ideal in $R$ contains a nonzerodivisor, then $R$ is a domain.

Proof. Consider the set $S$ of nonzerodivisors. It is a multiplicative subset of $R$. Hence any ideal maximal with respect to not intersecting $S$ is prime, see Lemma 10.27.8. Thus, if every nonzero prime ideal contains a nonzerodivisor, then $(0)$ is prime, i.e., $R$ is a domain. $\square$


Comments (4)

Comment #67 by Rankeya on

I think the first item in the formulation of the lemma is incorrect. The term zerodivisor should be replaced by nonzerodivisor.

Comment #68 by on

Haha, yes! Of course, since every ideal contains 0 the statement of part (1) wasn't wrong, it was just completely useless. Thanks!

Comment #3541 by Laurent Moret-Bailly on

In (2), should be assumed nonzero.


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