Jacobson radical and unit

Proof by contraposition: Assume there exists an \(y \in A\) such that \(1 - xy \in A \setminus A^{\times}\), then there exists a maximal ideal \(\mathfrak{m}\) with \(1 - xy \in \mathfrak{m}\) Furthermore \(xy \not\in \mathfrak{m}\), because otherwise \(1 \in \mathfrak{m}\). Hence \(x \not\in \mathfrak{m}\)

Assume there exists a maximal ideal with \(x \not\in \mathfrak{m}\), then

\begin{align*} x + \mathfrak{m} \neq 0 \in A/\mathfrak{m} \end{align*}

and because of the characterization of maximales Ideal eines kommutativen Rings und Faktorring als Körper we get that there exists a \(y \in A\) such that

\begin{align*} 0 + \mathfrak{m} = 1 - xy + \mathfrak{m} \Rightarrow 1 - xy \in \mathfrak{m} \end{align*}

Therefore it also holds, that

\begin{align*} 1 - xy \in \mathfrak{m} \subseteq A \setminus A^{\times} \end{align*}

(see: von einer Einheit erzeugtes Ideal und triviales Ideal)