An example is provided of a left semihereditary ring R with an idempotent ideal I such that R/I is not left semihereditary. Two related positive results are provided: a) a left semiartinian, left p.p. ring has nonsingular Loewy factors; b) if R is a commutative von Neumann regular ring and D is an idempotent ideal of R[X] then R[X]/D is semihereditary.