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From regular modules to von Neumann regular rings via coordinatization. (English) Zbl 1313.16021

Summary: We establish a very close link (in terms of von Neumann’s coordinatization) between regular modules introduced by Zelmanowitz, on one hand, and von Neumann regular rings, on the other hand: we prove that the lattice \(\mathcal L^{fg}(M)\) of all finitely generated submodules of a finitely generated regular module \(M\), over an arbitrary ring, can be coordinatized as the lattice of all principal right ideals of some von Neumann regular ring \(S\).

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D25 Ideals in associative algebras
06C20 Complemented modular lattices, continuous geometries
06D05 Structure and representation theory of distributive lattices
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