ON THE KRULL DIMENSION OF TRANSCENDENTAL EXTENSIONS OF COMMUTATIVE RINGS

摘要

IfRis a commutative ring with identity andt1,t2,#x2026;,tn, are transcendental overRsuch that no tiis algebraic over the ringRt1,#x2026;,t1-1,ti+1,#x2026;,tn it is proved that there exists a minimal prime idealPinRsuch that dimR+1#x2264; dimRt1,#x2026;,tn = dimRt1,#x2026;,tn/Pt1,#x2026;,tn. This equality is used to prove results for Rt1,#x2026;tnsimilar to those holding for the integral domain(R/P)t1,#x2026;,tn with 31,#x2026;t1,#x2026;,tnindeterminates overR/P.