Abstract.It is shown that if R is a commutative ring with identity and A is a multiplicatively closed set of finitely generated nonzero ideals of R , then the operation / -» IA = [JKeA(IK '■ K) is a closure operation on the set of ideals / of R that satisfies a partial cancellation law, and it is a prime operation if and only if R is A-closed. Also, if none of the ideals in A is contained in a minimal prime ideal, then IA Ç Ia , the integral closure of / in R , and if A is the set of all such finitely generated ideals and / contains an ideal in A , then IA = /" . Further, R has a natural A-closure RA , A -> AA is a closure operation on a large set of rings A that contain R as a subring, A -► AA behaves nicely under certain types of ring extension, and every integral extension overring of R is RA for an appropriate set A . Finally, if R is Noetherian, then the associated primes of IA are also associated primes of IAK and (IK)A for all K e A.