CATENARIAN PROPERTY OF MIXED POLYNOMIAL AND POWER SERIES RINGS OVER A PULLBACK

摘要

Let T be an integral domain with a maximal ideal M, φ: T → K：=T/M the natural surjection,and R the pullback φ~(-1) (D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[X_1]]… [x_n]] to be catenarian,where each [x_i]] is fixed as either [x_i] or [[x_i]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x_1]] …[x_n]]→ Spec（k[x_1]] …[x_n]]is a homeomovphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x_1]]…[x_n]] is catenarian.