Let ππL := Frm(π(β), L). We show that if L is a P-frame then ππL is an β΅0-self-injective ring. We prove that a zero-dimensional frame L is extremally disconnected if and only if ππL is a self-injective ring. Finally, it is shown that ππL is a Baer ring if and only if ππL is a continuous ring if and only if ππL is a complete ring if and only if ππL is a CS-ring.
Let C(L) be the ring of real-valued continuous functions on a frame L. In this paper, strongly fixed ideals and characterization of maximal ideals of C(L) which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of C(L), is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than spatiality and they are equivalent in the case of conjunctive frames. Assuming Axiom of Choice, compact frames are weakly spatial.2010 Mathematics Subject Classification: primary 06D22; secondary 13A15, 13C99. Key words and phrases: frame, ring of real-valued continuous functions, weakly spatial frame, fixed and strongly fixed ideal.
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