On rings in which every maximal one-sided ideal contains a maximal ideal

Given a ring R, consider the condition: (*) every maximal right ideal of R contains a maximal ideal of R We show that, for a ring R and 0 not equal e( 2) = e is an element of R such that ele not subset of or equal to eRe for e very proper ideal I of R, R satisfies (*) ii and only if eRe satisfies (*). Hence with the help of some other results, (*) is a Morita invariant prope rty. For a simple ring R, R[x] satisfies (*) if and only if R[x] is not rig ht primitive. By this result, if R is a division ring and R[x] satisfies (* ), then the Jacobson conjecture holds. We also show that for a finite centr alizing extension S of a ring R, R satisfies (*) if and only if S satisfies (*).