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We compute the monoid V (L K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L K (E) and the lattice of order-ideals of(More)
We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if R is a purely infinite simple ring, then K 0 (R) + = K 0 (R), the monoid of isomorphism classes of finitely generated projective R-modules is isomorphic to the monoid obtained(More)
Given an action α of a monoid T on a ring A by ring endomorphisms, and an Ore subset S of T , a general construction of a fractional skew monoid ring S op * α A * α T is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case S is a subsemigroup of a group G such that G = S −1 S, we obtain a G-graded ring S op * α A(More)
We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right R-modules over a von Neumann regular ring R. This survey consists of four sections. Section 1 introduces the realization problem(More)
Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB−rings. These constitute a considerable enlargement of the class of rings with stable rank one (B−rings), and include examples like End F (V), the ring of endomorphisms of a vector space V over some field F, and B(F), the ring of(More)
A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric(More)