The exchange property for purely infinite simple rings

We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then <TEX>$1_R$</TEX> = u + <TEX>$\upsilon$</TEX> with u, <TEX>$\upsilon$</TEX> <TEX>$\in$</TEX> U(R) if and only if for any a <TEX>$\in$</TEX> R, there exist u, <TEX>$\upsilon$</TEX> <TEX>$\in$</TEX> U(R) such that a = u + <TEX>$\upsilon$</TEX>. Furthermore, there exists u <TEX>$\in$</TEX> U(R) such that <TEX>$1_R\;{\pm}\;u\;\in\;U(R)$</TEX> if and only if for any a <TEX>$\in$</TEX> R, there exists u <TEX>$\in$</TEX> U(R) such that <TEX>$a\;{\pm}\;u\;\in\;U(R)$</TEX>. We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.

Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

We characterize exchange rings having stable range one. An exchange ring R has stable range one if and only if for any regular a ∈ R, there exist an e ∈ E(R) and a u ∈ U(R) such that a = e + u and aR ⋂ eR = 0 if and only if for any regular a ∈ R, there exist e ∈ r.ann(a +) and u ∈ U(R) such that a = e + u if and only if for any a, b ∈ R, R/aR ≅ R/bR ⇒ aR ≅ bR.

We find a distributive (v, 0, 1)-semilattice S of size $\\aleph_1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: - There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to S. - There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to S. These results are established by constructing an infinitary statement, denoted here by URPsr, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice S.

An ideal I of an exchange ring R is strongly separative provided that for any A,B ∈ FP(R), A⊕ A≅ A⊕ B ↠ A≅ B. If I is a strongly separative ideal of an exchange ring R, then each a ∈ R satisfying is unit-regular. The converse is true for regular ideals. Furthermore, we prove that strong separativity for such regular ideals can be determined only by one-sided units. Clean property of elements in strongly separative ideals of exchange rings is also studied.

Square-free modules with the exchange property

Stable modules and a theorem of Camillo and Yu

Let Mk be a right k-module with endomorphism ring E=End(Mk). We prove that if E is an Abelian exchange ring, then Mk has the full exchange property. We also give an extension of this result in the case E is regular.