Radicals of semiperfect rings related to idempotents

For a semiperfect ring $A$ we prove the existence of the minimal ideal $\mathcal M(A)$ (modular radical) such that the quotient ring $A/\mathcal M(A)$ has the identity element, and of the minimal ideal $\mathcal W(A)$ (Wedderburn radical) such that the quotient ring $A/\mathcal W(A)$ is decomposable into a direct sum of matrix rings over local rings. A simple criterion of such decomposability is given for left Noetherian semiperfect rings and left perfect rings.