Grid and peer-to-peer (P2P) networks are two ideal technologies for file sharing. A P2P grid is a special case of grid networks in which P2P communications are used for communication between nodes and trust management. Use of this technology allows creation of a network with greater distribution and scalability. Semantic grids have appeared as an expansion of grid networks in which rich resource metadata are revealed and clearly handled. In a semantic P2P grid, nodes are clustered into different groups based on the semantic similarities between their services. This paper proposes a reputation model for trust management in a semantic P2P Grid. We use fuzzy theory, in a trust overlay network named FR TRUST that models the network structure and the storage of reputation information. In fact we present a reputation collection and computation system for semantic P2P Grids. The system uses fuzzy theory to compute a peer trust level, which can be either: Low, Medium, or High. Our experimental results demonstrate that FR TRUST combines low (and therefore desirable) a good computational complexity with high ranking accuracy.