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 not left annihilated by $r$. Given any ring $R$, the quotient $R/J(R)$ is semiprimitive.
Some authors occasionally say Jacobson ideal.
(Jacobson radical of formal power series algebra)
The Jacobson radical of a formal power series algebra consists of those formal power series whose constant term vanishes.
(Jacobson radical of local ring)
The Jacobson radical of a local ring is the set of non-invertible elements.
(Jacobson radical of local prefield ring)
The Jacobson radical of a local prefield ring is the set of zero divisors.
(Jacobson radical of a possibly trivial local ring)
The Jacobson radical of a possibly trivial local ring $R$ is the set of elements $x \in R$ such that $x$ being invertible implies that $0 = 1$.
Named after Nathan Jacobson.
EoM: Jacobson radical
Wikipedia, Jacobson radical
Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) [doi:10.1007/978-94-017-9944-7, pdf]
Last revised on August 21, 2024 at 02:39:50. See the history of this page for a list of all contributions to it.