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== Congruence lattices of lattices and nonstable K-theory of von Neumann regular rings ==
We recall that for a (unital, associative) [[ring (mathematics)|ring]] ''R'', we denote by ''V(R)'' the (conical, commutative) monoid of isomorphism classes of finitely generated projective right ''R''-modules, see [[Refinement monoid|here]] for more details. Recall that if ''R'' is von [[Von Neumann regular ring|Neumann regular]], then ''V(R)'' is a [[refinement monoid]]. Denote by Id<sub>c</sub> ''R'' the (∨,0)-semilattice of finitely generated [[Ideal (ring theory)|two-sided ideals]] of ''R''. We denote by ''L(R)'' the lattice of all principal right ideals of a von Neumann regular ring ''R''. It is well-known that ''L(R)'' is a [[Complemented lattice|complemented]] [[
The following result was observed by Wehrung, building on earlier works mainly by Jónsson and Goodearl.<br /><br />
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