σ-Zero Sets and Characterization of σ-Strongly Fixed Maximal Ideals in RH for σ-Frames H

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 envelopeFH. Within this framework, we establish a one-to-one correspondence between σ-points of H and points of FH; the corresponding point in FH is termed the prime envelope of the σ-point. For each σ-point π, we define a maximal ideal Mπ via the function π˜, 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 Zσα is intrinsically linked to a classical zero set ZjHα in the frame envelope, which we call its zero envelope. We define σ-strongly fixed ideals and prove that the ideals Mπ provide a complete characterization of all σ-strongly fixed maximal ideals in RH. Collectively, these results establish a comprehensive point-free foundation for analyzing measurable function rings via σ-frames.