Polynomial and Power Series Rings with Finite Quotients

We determine the rings R with the property that the quotient ring R[X]/I (respectively, R[[X]]/I) is finite for every nonzero ideal I of the polynomial ring R[X] (respectively, of the power series ring R[[X]]). We also classify the rings R such that R[X]/I (R[[X]]/I) is a finite left R[X]- module (R[[X]]-module) for every nonzero left ideal I of R[X] (of R[[X]]).