LET X BE A TYCHONOFF SPACE, I BE AN IDEAL OF C (X) AND A (X) BE AN INTERMEDIATE ALGEBRAOF C (X). WE SHOW THAT UIX=UIZX=UIZX=UX È Q (I). MOREOVER, THESE EQUALITIES AREGENERALIZED TO THE IDEALS OF A (X) BY INVESTIGATING A TOPOLOGICAL CHARACTERIZATION FOR THE SMALLESTZ-IDEAL IN A (X) CONTAINING AN IDEAL J OF A (X), JZ(A). FINALLY, SOME PROPERTIES OF THE MAPPING U, WHICH ASSIGNS TO EACH SUBSET A OF C (X) THE RAELCOMPACT SPACE UAX, ARE GIVEN.