  In 1994, Matsuda and Okabe introduced the notion of semistar operation. This
concept extends the classical concept of star operation (cf. for instance,
Gilmer's book \cite{G}) and, hence, the related classical theory of ideal
systems based on the works by W. Krull, E. Noether, H. Pr\"{u}fer and P.
Lorenzen from 1930's. In \cite{FL1} and \cite{FL2} the current authors
investigated properties of the Kronecker function rings which arise from
arbitrary semistar operations on an integral domain $D$. In this paper we
extend that study and also generalize Kang's notion of a star Nagata ring
\cite{Kang:1987} and \cite{Kang:1989} to the semistar setting. Our principal
focuses are the similarities between the ideal structure of the Nagata and
Kronecker semistar rings and between the natural semistar operations that these
two types of function rings give rise to on $D$.
