## 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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### References:

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