Semi-irreducible Zariski spaces of modules

In this paper, we state conditions under which, the family of semi-irreducible submodules of a module determine a Zariski space of that module and study the properties of this space. Also we characterize semi-irreducible submodules of finitely generated modules over Dedekind domains. Moreover, assuming that M and M' are finitely generated modules over a Dedekind domain having isomorphic semi-irreducible Zariski spaces, we find some common properties of M and M'. In particular, we show that in this case the torsion-free components of M and M' have the same rank and the torsion submodules of M/N and M'/N' are isomorphic, where N and N' are the intersection of all semi-irreducible submodules of M and M', respectively.