Krull dimension, overrings and semistar operations of an integral domain

In this paper, we will present new developments in the study of the links between the cardinality of the sets O(R) of all overrings of R, SSFc(R) of all semistar operations of finite character when finite to the Krull dimension of an integral domain R. In particular, we prove that if |SSFc(R)|=n+dimR, then R has at most n−1 distinct maximal ideals. Moreover, R has exactly n−1 maximal ideals if and only if n=3. In this case R is a Prüfer domain with exactly two maximal ideals and Y-graph spectrum. We also give a complete characterizations for local domains R such that |SSFc(R)|=3+dimR, and nonlocal domains R with |SSFc(R)|=|O(R)|=n+dimR for n=4, n=5, n=6 and n=7. Examples to illustrate the scopes and limits of the results are constructed.