nLab Jacobson radical

Contents

Contents

Definition

Given a unital (typically noncommutative) ring RR, the Jacobson radical J(R)J(R) is defined as the set of elements rRr\in R satisfying the following equivalent properties:

  1. For each simple left RR-module MM, rM=0r M =0.
  2. Each maximal left ideal of RR contains rr.
  3. For all xRx\in R, 1rx1 - r x is left invertible in RR.

Alternatively,

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

The properties required remain the same if one interchanges left and right (modules, invertibility etc.) i.e. J(R)=J(R op)J(R)=J(R^{op}).

J(R)J(R) is a 22-sided ideal in RR. The rings for which J(R)=0J(R)=0 are called semiprimitive rings. In other words, for each nonzero element rr in a semiprimitive ring, by the definition, there is a simple module not left annihilated by rr. Given any ring RR, the quotient R/J(R)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 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 RR is the set of elements xRx \in R such that xx being invertible implies that 0=10 = 1.

References

Last revised on January 22, 2023 at 11:09:44. See the history of this page for a list of all contributions to it.