Exchange rings in which all regular elements are one-sided unit-regular

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.

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.

It is proven that every purely infinite simple ring is an exchange ring. This result is applied to determine those Leavitt algebras that are exchange rings.

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.

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.

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.

Stable modules and a theorem of Camillo and Yu

A ring R has weakly stable range one provided that aR+bR=R implies that there exists a <TEX>$y{\in}R$</TEX> such that <TEX>$a+by{\in}R$</TEX> is right or left invertible. We prove, in this paper, that every regular element in an exchange ring having weakly stable range one is the sum of an idempotent and a weak unit. This generalize the corresponding result of one-sided unit-regular ring. Extensions of power comparability and power cancellation are also studied.

Let A and B be generalized stable rings. It is shown that every von Neumann regular element of a Morita context is the product of an idempotent and one-sided units. Also we show that such Morita contexts are GE-rings.