Jacobson radical (original) (raw)

The following are alternative characterizations of the Jacobson radical J⁢(R):

1. 1. The intersection of all left primitive ideals.
2. 1. The intersection of all maximal left ideals.
3. 1. The set of all t∈R such that for all r∈R, 1-r⁢t isleft invertible (i.e. there exists u such that u⁢(1-r⁢t)=1).
4. 1. The largest ideal I such that for all v∈I, 1-v is a unit in R.
5. 1. (1) - (3) with “left” replaced by “right” and r⁢t replaced by t⁢r.

The Jacobson radical can also be defined for non-unital rings. To do this, we first define a binary operation ∘ on the ring Rby x∘y=x+y-x⁢y for all x,y∈R. Then (R,∘) is a monoid, and the Jacobson radical is defined to be the largest ideal I of Rsuch that (I,∘) is a group. If R is unital, this is equivalent to the definitions given earlier.