In this paper, for eachlattice-VALUED map A®L with some properties, a RING representation A®RL is constructed. This representation is denoted byc which is an f-RING homomorphism and a Q-linear map, where its indexc, mentions to a lattice-VALUED map. We use the notation a dpqa= (a − p)+^ (q − a)+, where p, q Î Q and a Î A, that is nominated asinterval projection. To get a well-defined f-RING homomorphism tc, we need such concepts asbounded, continuous, and Q-compatible for c, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-VALUED map c f: A®L for each f-RING homomorphism f: A®RL. It is proved that crc=cr and tcf=f, which they make a kind of correspondence relation between RING representations A®RL and the lattice-VALUED maps A®L, where the mapping cr: A®L is called a REALizationof c. It is shown that tcr=tc and crr=cr.