Resumen de When is a pullback a locally divided domain?

David E. Dobbs

• Let P be a property of some (commutative integral) domains such that: if A is a proper subring of a domain B such that Spec(A)= Spec(B) as sets, then A has P if and only if B has P; if A is a domain and Q is a prime ideal of A, then the CPI-extension of A with respect to Q has P if and only if both A/Q and the localization of A at Q have P; and if A is a domain, then A has P if and only if the localization of A at M has P for each maximal ideal M of A. Examples of such P include the property of being a going-down domain and the property of being a locally divided domain. Let T be a domain, Q a maximal ideal of T, p the canonical surjection from T to T/Q, D a subring of T/Q, and R the inverse image of D under p. Then the pullback R (of p and the inclusion map from D to T/Q) has P if and only if both T and D have P. By suitably modifying the above requirements on the property P, we obtain a companion result which applies, in particular, when P is the property of being a locally pseudo-valuation domain.