A Note on Lifting of Chains of Prime Ideals

In [3, Theorem 3] it is proved that for a ring homomorphism : → such that () = for all ideals of , given any chain of prime ideals ℘ ⊆ ℘ ⊆ … ℘ in there exists a chain of prime ideals ⊆ ⊆ … in such that () = ℘. Here, under the weaker assumption (℘) = ℘ for all prime ideals ℘ of , we give a necessary and sufficient condition for validity of [3, Theorem 3] and deduce the theorem as a corollary. Further, we prove a general version of [2, Lemma 6 (iii)] and deduce that for any two faithfully flat R-algebras , Krull dimension of ⊗ is bigger than equal to the sum of Krull dimensions of and , a result proved in [2] when R is a field