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

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