a ring modulo its Jacobson radical is semiprimitive
Let be a ring. Then .
Proof:
We will only prove this in the case where is a unital ring(although it is true without this assumption).
Let .By one of the characterizations of the Jacobson radical
, is left invertible for all ,so there exists such that .
Then for some .There is a such that ,and we have .
Since this holds for all ,it follows that , and therefore .