On r-ideals of R(L)

Abstract

In this paper, we study the concept of r-ideal (a proper ideal I in a ring R is said to be an
r-ideal if ra 2 I with Ann(r) = (0), implies that a 2 I for each a; r 2 R) in the ring R(L), as the point-free
counterpart of C(X) and a reduced commutative ring. We investigate the behavior of this type of ideal in the cozero complemented frames, P-frames, almost P-frames, and weakly almost P-frames. We prove the characterization of these frames via the concept of r-ideal in the ring R(L).

We examine other groups of ideals, namely zr-ideal and sr-ideal in the ring R(L), by combining the concept of r-ideal with z-ideal and also with the semiprime ideal. We show that the sum of the zr-ideals in the ring R(L) has the same behavior as the z0-ideals in this ring in a simple way: The sum of every two
zr-ideals in R(L) is a zr-ideal or all of R(L) if and only if L is a quasi-F-frame. Here, this fact is also proved for sr-ideals. ideals We provide.