It is shown that if R R is a commutative ring with identity and Δ \Delta is a multiplicatively closed set of finitely generated nonzero ideals of R R , then the operation I → I Δ = ∪ K ∈ Δ ( I K : K ) I \to {I_\Delta } = { \cup _{K \in \Delta }}(IK:K) is a closure operation on the set of ideals I I of R R that satisfies a partial cancellation law, and it is a prime operation if and only if R R is Δ \Delta -closed. Also, if none of the ideals in Δ \Delta is contained in a minimal prime ideal, then I Δ ⊆ I a {I_\Delta } \subseteq {I_a} , the integral closure of I I in R R , and if Δ \Delta is the set of all such finitely generated ideals and I I contains an ideal in Δ \Delta , then I Δ = I a {I_\Delta } = {I_a} . Further, R R has a natural Δ \Delta -closure R Δ , A → A Δ {R^\Delta },A \to {A^\Delta } is a closure operation on a large set of rings A A that contain R R as a subring, A → A Δ A \to {A^\Delta } behaves nicely under certain types of ring extension, and every integral extension overring of R R is R Δ {R^\Delta } for an appropriate set Δ \Delta . Finally, if R R is Noetherian, then the associated primes of I Δ {I_\Delta } are also associated primes of I Δ K {I_\Delta }K and ( I K ) Δ {(IK)_\Delta } for all K ∈ Δ K \in \Delta .