Forcing linearity numbers for modules over rings with nontrivial idempotents

We determine forcing linearity numbers of modules V over rings R with nontrivial idempotents. If 1=e1+ +en, n 2, where the ei are orthogonal idempotents, we determine the forcing linearity number of V in terms of certain modules over the rings eiRei. Thus if each eiRei is a division ring we obtain precise answers. Also, as a consequence, we find the forcing linearity numbers for all modules over Artinian rings.