Rings of frame maps from P(R) to frames which vanish at infinity

Let FP(L) be the set of all frame maps from P(R) to L, which is an f-ring. In this paper, we introduce the subrings FP1(L) of all frame maps from P(R) to L which vanish at infinity and FPK(L) of all frame maps from P(R) to L with compact support. We prove FP1(L) is a subring of FP(L) that may not be an ideal of FP(L) in general and we obtain necessary and sufficient conditions for FP1(L) to be an ideal of FP(L). Also, we show that FPK(L) is an ideal of FP(L) and it is a regular ring. For f 2 FP(L), we obtain a sufficient condition for f to be an element of FP1(L) (FPK(L)). Next, we give necessary and sufficient conditions for a frame to be compact. We introduce FP-pseudocompact and next we establish equivalent condition for an FP-pseudocompact frame to be a compact frame. Finally, we study when for some frame L with FP1(L) 6= (0), there is a locally compact frame M such that FP1(L) = FP1(M) and FPK(L) = FPK(M).