On analogues involving zero-divisors of a domain-theoretic result of Ayache

Ayache has recently proved that if R is an integrally closed domain such that each overring of R is treed, then R is a locally pseudo-valuation domain. We investigate the extent to which the analogue (in which one concludes that R is a locally pseudo-valuation ring) holds if R is generalized to a commutative ring. A positive result is obtained if R is an idealization D(+)K where D is an integrally closed LPVD each of whose overrings is treed (for instance, a Prüfer domain) with quotient field K. However, the analogue fails in general for (quasi-local) idealizations of the form R = A(+)A where A = Z/pZ with p a prime number and n ≥ 3. A positive result for the analogue is obtained for certain reduced rings R (namely, weak Baer rings that are integrally closed), but an example using the A+B construction shows that the analogue fails in general for reduced rings that are total quotient rings.