Jacobson Radical, Maximal Ideals in an Artinian Ring
Maximal Ideals in an Artinian Ring
Let R be left artinian,
and let J be its jacobson radical.
The maximal ideals of R and the maximal ideals of R/J correspond.
Mod out by J and look at the quotient ring.
If R is jacobson semisimple and left artinian then it
is semisimple.
The maximal left ideals of a semisimple ring have been
characterized.
If R is commutative and artinian, there are finitely many maximal ideals.