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We study the behaviour of the finiteness of the set of associated primes oflocal cohomology modules, more generally of Lyubeznik functors, under variousring extensions. At first, we review the results for flat and faithfully flatextensions and we present new applications of them. Then, we focus how thefiniteness property of the set of associated primes of local cohomology modulesand Lyubeznik functors is transferred from extended ring to the base ring ofpure and cyclically pure ring extensions. We show that finiteness property canbe transferred from a ring to its pure local subring and this extends theresult of Theorem 1.1 of \cite{Nu}. Further, we observed that under mildconditions on the rings, finiteness property comes down from cyclically purering extensions to its local base ring. In particular, we observe that the setof associated primes of Lyubeznik functors of a cyclically pure local subring(which turns out to be Cohen-Macaulay) of equicharacteristic or unramifiedregular local ring, is finite. There is an appendix on behaviour of the Bassnumbers under pure and cyclically pure ring extensions.

Behaviour of Finiteness of the Set of Associated Primes under Ring Extensions