Extension of Semistar Operations

Let R C T be an extension of integral domains and * be a semistar operation stable of finite type on R. We define a semistar operation *1 on T in the following way: for each nonzero T-submodule E of the quotient field K1 of T, let E*1 = U{E :K1 JT I J E F*}, where K1 denotes the quotient field of T and F* the localizing system associated to *. In this paper we investigate the basic properties of *1. Moreover, we show that the map phi which associates to a semistar operation * stable and of finite type on R, the semistar operation *1 is continuous. Furthermore, we give sufficient conditions for phi to be a homeomorphism. 2020 MSC. 13A15, 13B02, 13G05