Abstract :
In (J. Pure Appl. Algebra 179 (2003) 1) it is proved that for a ring homomorphism α : R→A such that α−1(IA)=I for all ideals I of R, given any chain of prime ideals Weierstrass p0subset of or equal toWeierstrass p1subset of or equal tocdots, three dots, centeredsubset of or equal toWeierstrass pn in R there exists a chain of prime ideals q0subset of or equal toq1subset of or equal tocdots, three dots, centeredsubset of or equal toqn in A such that α−1(qi)=Weierstrass pi. Here, under the weaker assumption α−1(Weierstrass pA)=Weierstrass p for all prime ideals Weierstrass p of R, we give a necessary and sufficient condition for validity of (J. Pure Appl. Algebra 179 (2003) 1) and deduce the theorem as a corollary. We note that α−1(IA)=I for all ideals I in R is not a necessary condition.
