Ascent and descent of going-down rings for integral extensions

The author and Ira J. Papick have termed an integral domainRa going-down ring ifR⊂Tsatisfies going-down for each domainTcontainingR. The present paper investigates conditions which, for an integral extensionA⊂Bof domains, imply thatA(respectivelyB) is going-down wheneverB(respectivelyA) is going-down. This explains the “descent” (respectively “ascent”) in the title. Two typical results (the first about descent, the second about ascent) are given next.