This paper investigates the fundamental properties of function rings over σ ‐frames, focusing on their maximal ideals. To bridge the study of σ ‐frames with classical frame theory, we systematically employ the construction of the frame envelope . Within this framework, we establish a one‐to‐one correspondence between σ ‐points of H and points of ; the corresponding point in is termed the prime envelope of the σ ‐point. For each σ ‐point π , we define a maximal ideal M π via the function , which is proved to be a surjective f ‐ring homomorphism and plays a pivotal role in our work. We introduce σ ‐zero sets as a key technical tool and show that each σ ‐zero set Z σ ( α ) is intrinsically linked to a classical zero set Z ( j H α ) in the frame envelope, which we call its zero envelope . We define σ ‐strongly fixed ideals and prove that the ideals M π provide a complete characterization of all σ ‐strongly fixed maximal ideals in . Collectively, these results establish a comprehensive point‐free foundation for analyzing measurable function rings via σ ‐frames.
Nejad et al. (Thu,) studied this question.