Characterisation of Jacobson Radical

Theorem
Let $$A$$ be a commutative ring.

Let $$A^\times$$ be the group of units of $$A$$

Let $$\text{Jac}(A)$$ be the Jacobson radical of $$A$$.

Then


 * $$\text{Jac}(A)=\left\{a\in A:1-ax\in A^\times\text{ for all }x\in A\right\}$$

Proof
First suppose that $$1-xy\notin A^\times$$.

Then it is contained in some maximal ideal $$\mathfrak m\subseteq A$$.

Then $$x\in \text{Jac}(A)\subseteq\mathfrak m$$ implies that $$y\in\mathfrak m$$ and therefore $$1\in \mathfrak m$$, which is impossible.

This shows that $$\text{Jac}(A)\subseteq\left\{a\in A:1-ax\in A^\times\text{ for all }x\in A\right\}$$.

Now suppose that $$x\notin \mathfrak m$$ for some maximal ideal $$\mathfrak m$$ of $$A$$.

Since $$\mathfrak m$$ is maximal, $$x$$ and $$\mathfrak m$$ generate $$A$$.

Therefore there exist $$w\in\mathfrak m$$ and $$y\in A$$ such that $$w+xy=1$$.

Thus $$1-xy\in\mathfrak m$$, and $$1-xy\notin A^\times$$.

This completes the proof.