A note on lifting of chains of prime ideals

AbstractIn (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 ℘0⊆℘1⊆⋯⊆℘n in R there exists a chain of prime ideals q0⊆q1⊆⋯⊆qn in A such that α−1(qi)=℘i. Here, under the weaker assumption α−1(℘A)=℘ for all prime ideals ℘ 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