Exchange rings with stable range one

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.

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.

A short, elementary proof is given that right exchange rings are left exchange, with an application to exchange rings with one in the stable range.

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.

A ring R is defined to be weakly normal if for all a, r \\in R and e \\in E(R), ae = 0 implies Rera is a nil left ideal of R, where E(R) stands for the set of all idempotent elements of R. It is proved that R is weakly normal if and only if Rer(1-e) is a nil left ideal of R for each e \\in E(R) and r \\in R if and only if Tn(R, R) is weakly normal for any positive integer n. And it follows that for a weakly normal ring R (1) R is Abelian if and only if R is strongly left idempotent reflexive; (2) R is reduced if and only if R is n-regular; (3) R is strongly regular if and only if R is regular; (4) R is clean if and only if R is exchange. (5) exchange rings have stable range 1.

Stable modules and a theorem of Camillo and Yu

We investigate partial cancellation of modules and show that if an ideal I of an exchange ring R has stable range one, then A⊕B≌A⊕C implies B≌C for all A∈FP (I). The converse is true when R is a regular ring. For an ideal I of a regular ring, we also show that I has stable range one if and only if perspectivity is transitive in L(A) for all A∈ FP (I). These give nontrivial generalizations for unit-regularity.

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.