For a domain D, D is a
Prüfer domain if and only if D∕P is a Prüfer domain for every prime ideal P of D.
The same result does not hold for rings with zero divisors. In this paper it is shown
that for a Prüfer ring R with prime ideal P, R∕P is a Prüfer ring if P is not
properly contained in an ideal consisting entirely of zero divisors. An example is
provided to show that, in general, this is the best possible result. According to M.
Boisen and P. Sheldon, a pre-Prüfer ring is defined to be a ring for which every
proper homomorphic image is a Prüfer ring. In this paper it is proved that
for a pre-Prüfer ring R containing zero divisors, the integral closure of
R is a Prüfer ring. Furthermore, if R is a reduced pre-Prüfer ring with
more than two minimal prime ideals, then R is already integrally closed and,
moreover, R is not only Prüfer but arithmetical as well. An example is
provided of an integrally closed pre-Prüfer domain which is not a Prüfer
domain.
