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Zariski spaces of modules over arbitrary rings. (English) Zbl 1168.16027

Summary: It is proved that if \(M\) and \(M'\) are modules over an arbitrary ring \(R\), then the Zariski spaces \(\zeta(M)\) and \(\zeta(M')\) 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 \(\mathbf{RAD}(M)\) and \(\mathbf{RAD}(M')\) 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.

MSC:

16Y60 Semirings
16D80 Other classes of modules and ideals in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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