Semistar-operations of finite character on integral domains

In this paper, we investigate the semistar-operations of finite character on integral domains. We state a conditions under which the semistar-operation defined by a family of overrings of a domain R is of finite character. This notion leads us to give a new characterization of Prüfer domains and characterize Prüfer and Noetherian domains R for which each semistar-operation is of finite character. It turns out that R must be conducive (so local and one-dimensional) in the Noetherian case and conducive and each overring of R is divisorial for the Prüfer case. We also show that image for each nonlocal domain R and we characterize domains for which the equality holds.