Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras

The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea

Pseudoneoplastic lesions of the testis and paratesticular structures

Pseudotumors or tumor-like proliferations (non-neoplastic masses) and benign mimickers (non-neoplastic cellular proliferations) are rare in the testis and paratesticular structures. Clinically, these lesions (cysts, ectopic tissues, and vascular, inflammatory, or hyperplastic lesions) are of great interest for the reason that, because of the topography, they may be relevant as differential diagnoses. The purpose of this paper is to present an overview of the pseudoneoplasic entities arising in the testis and paratesticular structures; emphasis is placed on how the practicing pathologist may distinguish benign mimickers and pseudotumors from true neoplasia. These lesions can be classified as macroscopic or microscopic mimickers of neoplasia

Direct-sum behavior of modules over one-dimensional rings

Let R be a reduced, one-dimensional Noetherian local ring whose integral closure is finitely generated over R. Since is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated-modules are easily described, and every finitely generated-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules. © 2011 Springer Science+Business Media, LLC