  We show that a generalization of the theory of quantales and 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 quantales and prequantales. Applications include
representation theorems for precoherent prequantales and multiplicative
semilattices; characterizations of simple commutative quantales and simple
multiplicative lattices; 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 two potentially useful definitions of tight closure for
non-Noetherian commutative rings of prime characteristic.
