Semihereditary rings and FP-injective modules. (Q1407381)

The greater part of this paper concerns Rickartian rings (\(R\) is right Rickartian if all its principal right ideals are projective). There are three main theorems, each proved by means of a series of lemmas. The first of these proves that for an invariant right or left Rickartian ring \(A\), \(A\) being semihereditary is equivalent to every divisible right \(A\)-module being FP-injective. If \(A\) is either a distributive right semihereditary ring or a normal right Rickartian right Bézout ring, then the concepts of divisibility, 1-injectivity, finite injectivity and FP-injectivity are shown to be equivalent for a right \(A\)-module \(M\). The final theorem finds an equivalent condition for a right hereditary ring to be right distributive.