symmetric monoidal (∞,1)-category of spectra
Given a unital (typically noncommutative) ring $R$, the Jacobson radical $J(R)$ is defined as the set of elements $r\in R$ satisfying the following equivalent properties:
Alternatively,
The properties required remain the same if one interchanges left and right (modules, invertibility etc.) i.e. $J(R)=J(R^{op})$.
$J(R)$ is a $2$-sided ideal in $R$. The rings for which $J(R)=0$ are called semiprimitive rings. In other words, for each nonzero element $r$ in a semiprimitive ring, by the definition, there is a simple module left annihilated by $r$. Given any ring $R$, the quotient $R/J(R)$ is semiprimitive.
(Jacoboson radical of formal power series algebra)
The Jacobson radical of a formal power series algebra consist of those formal power series whose constant term vanishes.
See also