Let R be a commutative ring with identity,M be an R-module,(L)(M)denote the set of all submodules of M and(G)(€)(L)(M)\{OM}.For any submodule N of M,we set(G)Vd(N)={K ∈(G):K(∈) N} and(G)ζd(M)={(G)Vd(N):N ∈(L)(M)}.Consider x ⊆(L)(R)\{R},where(L)(R)is the set of all ideals of R.We set xV(I)={J∈x:I⊆J}and xζ(R)={xV(I):I ∈(L)(R)} for any ideal I of R.In this paper,we investigate when,for arbitrary x and(G)as above,xζ(R)and(G)ζd(M)form a topology and a semimodule,respectively.We investigate the structure of(G)ζd(M)in the case that it is a semimodule.
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