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

Abstract

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 ≠ e ² = e ∈ R such that ele ⫋ eRe every proper ideal I of R R satisfies (^*) if and only if eRe satisfies (^*). Hence with the help of some other results, (^*) is a Morita invariant property. For a simple ring R R[x] satisfies (^*) if and only if R[x] is not right 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 centralizing extension S of a ring R R satisfies (^*) if and only if S satisfies (^*).