nLab Jacobson radical

Definition

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:

1. For each simple left $R$-module $M$, $r M =0$.
2. Each maximal left ideal of $R$ contains $r$.
3. For all $x\in R$, $r x$ is left quasiregular, i.e. $1 - r x$ is left invertible in $R$

Alternatively,

1. $J(R)$ is the intersection of all maximal left ideals of $R$.
2. $J(R)$ is the intersection of all maximal right ideals of $R$.

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.

Examples

Example

(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.

Example

(Jacobson radical of local ring)

The Jacobson radical of a local ring is the set of non-invertible elements.

Example

(Jacobson radical of the trivial ring)

The Jacobson radical of a trivial ring $R$ is the trivial ideal $0 R$.

Example

(Jacobson radical of local prefield ring)

The Jacobson radical of a local prefield ring is the set of zero divisors.

Example

(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 $0 = 1$.

Related concepts

• radical
• scale
• Nakayama lemma
• quasiregular element

References

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]