I know that the Jacobson radical of a (non-commutative) rng $R$ is the intersection of all annihilators of right irreducible modules. I know also that one can replace "right" with "left" in the previous definition. What if we consider the intersection of all annihilators of irreducible bimodules? Is this still the Jacobson radical of $R$?
By the annihilator of a bimodule $M$ I mean the largest ideal $A$ of $R$ such that $MA=0$ and $AM=0$.