Free ideals and real ideals of the ring of frame maps from P(R) to a frame

Let FP(L) (F∗P(L)) be the f-rings of all (bounded) frame maps from P(R) to a frame L. FP∞(L) is the family of all f∈FP(L) such that ↑f(−1n,1n) is compact for any n∈N and the subring FPK(L) is the family of all f∈FP(L) such that coz(f) is compact. We introduce and study the concept of real ideals in FP(L) and F∗P(L). We show that every maximal ideal of F∗P(L) is real, and also we study the relation between the conditions ``L is compact" and ``every maximal ideal of FP(L) is real''. We prove that for every nonzero real Riesz map φ:FP(L)→R, there is an element p in ΣL such that φ=pcoz˜ if L is a zero-dimensional frame for which B(L) is a sub-σ-frame of L and every maximal ideal of FP(L) is real. We show that FP∞(L) is equal to the intersection of all free maximal ideals of F∗P(L) if B(L) is a sub-σ-frame of a zero-dimensional frame L and also, FPK(L) is equal to the intersection of all free ideals FP(L) (resp., F∗P(L)) if L is a zero-dimensional frame. Also, we study free ideals and fixed ideals of FP∞(L) and FPK(L).