We show that the theory of nuclei on prequantales provides a noncommutative
and nonassociative abstract ideal theoretic setting for the theories of star
operations, semistar operations, semiprime operations, ideal systems, and
module systems, and conversely the latter theories motivate new results in the
theory of nuclei. Applications include a representation theorem for precoherent
prequantales; a construction of the largest finitary nucleus smaller than a
given nucleus on a precoherent prequantale; a construction of the smallest
semistar operation extending a given star operation on an integral domain; and
a definition of tight closure for non-Noetherian commutative rings of prime
characteristic.