= Unit criterion for the Jacobson radical
{title2=$x\in J(R)\iff 1-axb\in R^\times\text{ for all }a,b\in R$}
The <Jacobson radical> is the intersection of the ideals $\operatorname{Ann}_R(S)$ over all <simple modules>, where $\operatorname{Ann}$ is the <annihilator of a module>, so it is a two-sided <ideal>. If $y\in J(R)$, the <left ideal> $R(1-y)$ cannot be proper, since a containing maximal left ideal would also contain $y$ and hence $1$. Thus $c(1-y)=1$ for some $c$. Now $c=1+cy$ also differs from $1$ by a radical element and has a left inverse $d$. Multiplying $c(1-y)=1$ by $d$ gives $d=1-y$, and hence $(1-y)c=1$. This proves the unit assertion. Conversely, if $x$ is outside a maximal left ideal $L$, then $ax+l=1$ for some $a\in R$, $l\in L$. Consequently $1-ax\in L$ cannot be a <unit>, proving the reverse direction.
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