Ceken, Secil2024-06-122024-06-1220231005-38670219-1733https://doi.org/10.1142/S1005386723000445https://hdl.handle.net/20.500.14551/20618Let R be a commutative ring with identity, M be an R-module, L (M ) denote the set of all submodules of M and G subset of L ( M) \ { 0(M) } . For any submodule N of M, we set GV(d) ( N) = { K is an element of G : K subset of N } and G zeta(d) (M ) = { GV(d) ( N) : N is an element of L (M ) } . Consider chi subset of L ( R) \ { R } , where L (R ) is the set of all ideals of R. We set chi V (I ) = { J is an element of chi : I subset of J } and chi zeta (R ) = { chi V (I ) : I is an element of L (R ) } for any ideal I of R. In this paper, we investigate when, for arbitrary chi and G as above, chi zeta (R ) and G zeta(d) (M ) form a topology and a semimodule, respectively. We investigate the structure of G zeta(d) (M ) in the case that it is a semimodule.en10.1142/S1005386723000445info:eu-repo/semantics/closedAccessDual Zariski SemimoduleChi-Zariski SemiringCenter Dot-Coprime SubmoduleSubtractive SubspaceArticle304569584N/AWOS:0011155205000072-s2.0-85179784156Q3