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

Document Type : Research Paper

Authors

1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Postal Code 9617976487, Sabzevar, Iran

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

Abstract

Let FP(L) (FP(L)) be   the f-rings   of all (bounded) frame maps from P(R) to a frame L. FP(L) is  the family of all fFP(L) such that  f(1n,1n) is compact for any nN and the subring  FPK(L) is the family of all   fFP(L) such that coz(f) is compact. We  introduce  and study  the concept of   real ideals in FP(L) and FP(L). We  show  that every maximal ideal of FP(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 FP(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.,  FP(L)) if L is a zero-dimensional frame.  Also, we study free ideals and fixed ideals of    FP(L) and  FPK(L).

Keywords


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