Some new characterizations of finite frames and F-frames

Let R(L)R(L) stand for the ring of all real valued continuous functions on a completely regular frame L. We first show that if L   is generated by the cozeros of functions in RΨ(L)RΨ(L), the ring of all functions in R(L)R(L) which have pseudocompact support, then L   is finite if and only if RΨ(L)RΨ(L) is a Noetherian ring. We next show for a supportively normal frame L  , which in addition is also generated by the cozeros of functions in RΦ(L)RΦ(L), the ring of all functions in R(L)R(L) having realcompact support, that L   is finite if and only if RΦ(L)RΦ(L) is Noetherian. We further check that a frame (resp., a completely regular frame) L   is finite if and only if R(L)R(L) is a Noetherian/Artinian/semisimple/hereditary ring if and only if each maximal ideal of R(L)R(L) is principal. This last result expands one of the basic theorems of the present authors already achieved. In the next stage we show that L is an F  -frame if and only if each ideal of R(L)R(L) is flat if and only if for each f∈R(L)f∈R(L), pos(f)pos(f) and neg(f)neg(f) are completely separated. If PP is an ideal of closed quotients of L  , then a normal PP-continuous frame turns out to be an F  -frame when and only when for each f∈RP(L)f∈RP(L), the functions in R(L)R(L) having their support on PP, pos(f)pos(f) and neg(f)neg(f) are completely separated. Finally we realise that L is a P  -frame if and only if each prime ideal of R(L)R(L) is a z  -ideal if and only if every R(L)R(L)-module is flat and also that L   is basically disconnected if and only if R(L)R(L) is a semihereditary ring. It is worth mentioning that most of the above results, barring a few have their classical counterparts already available in the literature.