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Zariski Spaces of Modules over Arbitrary Rings

It is proved that if M and M′ are modules over an arbitrary ring R, then the Zariski spaces ζ(M) and ζ(M′) of varieties of submodules of M and M′, respectively, are isomorphic as semimodules over the Zariski semiring of varieties of ideals of R if and only if the R-lattices and of radical submodules of M and M′, respectively, are isomorphic. In this case, it is shown that although the modules M and M′ need not be isomorphic, they do have a number of properties in common.