Show that every non-unit of \(R\) is contained in a maximal ideal.
Solution
This follows because if \(x\in R\sm \unitsof{R}\), then \(Rx \normal R\) and \(Rx\neq R\) implies \(R/Rx \neq 0\). Then there exists some \(\bar \mfm \in \mspec R/Rx\), and by the correspondence theorem this lifts to some \(\mfm \in \mspec R\) containing \(Rx\).