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A dual Zariski topology for modules. (English) Zbl 1226.16006

A module \(M\) is called duo if every submodule is fully invariant. For every duo module \(M\) is defined the Zariski topology \(\tau^{fc}(M)\) on the set \(\text{Spec}^{fc}(M)\) of all fully coprime submodules of \(M\).
In the paper are studied interrelations between algebraic properties of a duo module \(M\) and topological properties of the topological space \(Z^{fc}(M)=(\text{Spec}^{fc}(M),\tau^{fc}(M))\). There are given conditions under which \(Z^{fc}(M)\) is: Noetherian, Artinian, irreducible, discrete, countably compact, compact, ultraconnected, connected.

MSC:

16D80 Other classes of modules and ideals in associative algebras
16N60 Prime and semiprime associative rings
16W80 Topological and ordered rings and modules
54H13 Topological fields, rings, etc. (topological aspects)
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C13 Other special types of modules and ideals in commutative rings
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