On P- frames and their generalisations

Abstract

In this dissertation, we study P -frames and their generalisations. On the generalisations of P - frames we consider, in particular essential P -frames, CP -frames, almost P -frames, F -frames, F ′-frames and PF -frames. We show that a frame L is a P -frame if and only if every ideal of RL is a z-ideal. We also consider R-modules and then show that a frame L is a P -frame if and only if every RL-module is flat. Furthermore, we consider the Artin-Rees property and show that a frame L is a P -frame if and only if RL is an Artin-Rees ring. Concerning CP -frames we show, analogously to P -frames, that a frame L is a CP -frame if and only if every ideal of RcL is a zc-ideal. It turns out that in CP -frame radical ideals are precisely zc-ideals. We show, regarding F -frames, that L is an F -frame if and only if RL is a B´ezout ring. We show that L is an F -frame if and only if every ideal of RL is convex. Finally, we introduce PF -frames and show that L is a PF -frame if and only if it is an essential P -frame which is also an F -frame.