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, SSF c (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 | SSF c (R) | = n + dim R, 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 | SSF c (R) | = 3 + dim R, and nonlocal domains R with | SSF c (R) | = | O (R) | = n + dim R for n = 4, n = 5, n = 6 and n = 7. Examples to illustrate the scopes and limits of the results are constructed.