On self-injectivity of the f-ring Frm(𝓟(ℝ), L)

Abstract

Abstract Let 𝓕𝓟L := Frm(𝓟(ℝ), L). We show that if L is a P-frame then 𝓕𝓟L is an ℵ0-self-injective ring. We prove that a zero-dimensional frame L is extremally disconnected if and only if 𝓕𝓟L is a self-injective ring. Finally, it is shown that 𝓕𝓟L is a Baer ring if and only if 𝓕𝓟L is a continuous ring if and only if 𝓕𝓟L is a complete ring if and only if 𝓕𝓟L is a CS-ring.