Question

Show that rad R is the intersection A of the annihilators of all simple left R-modules.

Answer 1

We show rad R = A. For any modular maximal left ideal m,we have (rad R) R ⊆rad R ⊆m. Therefore, (rad R) · R/m = 0, sorad R ⊆ A. Conversely, if a ∈ A, then aR ⊆m for any modular maximal left ideal, so aR ⊆rad R. Since rad R is quasi-regular,so is aR; hence a ∈rad R.