Let \(R\) be a commutative ring. An element \(a\in R\) is quasiregular if \(1-a\) is a unit of \(R\). The Jacobson radical \(J(R)\) is the largest ideal of \(R\) all of whose elements are quasiregular: every element of \(J(R)\) is quasiregular, and every ideal consisting of quasiregular elements is contained in \(J(R)\).
Let \(x\in J(R)\). If \(1-x\) were not a unit, it would lie in a maximal ideal \(\mfm\) by FF-QELG7, and so would \(x\), giving \(1\in\mfm\).
Let \(I\) be an ideal consisting of quasiregular elements, let \(x\in I\), and suppose \(x\notin\mfm\) for a maximal ideal \(\mfm\). Then \(\mfm+(x)=R\), so \(1=y+rx\) with \(y\in\mfm\) and \(r\in R\). Since \(rx\in I\), the element \(y=1-rx\) is a unit, contradicting \(y\in\mfm\). So \(I\subseteq\mfm\) for every maximal ideal \(\mfm\), that is, \(I\subseteq J(R)\).