Pere Ara

Separative cancellation for projective modules over exchange rings

A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative.

Strongly -regular rings have stable range one

A ring R is said to be strongly π-regular if for every a ∈ R there exist a positive integer n and b ∈ R such that an = an+1b. For example, all algebraic algebras over a field are strongly π-regular. We prove that every strongly π-regular ring has stable range one. The stable range one condition is especially interesting because of Evans’ Theorem, which states that a module M cancels from direct sums whenever EndR(M) has stable range one. As a consequence of our main result and Evans’ Theorem, modules satisfying Fitting’s Lemma cancel from direct sums. Introduction Let R be a ring, associative with unity. Recall that R has stable range one provided that, for any a, b ∈ R with aR + bR = R, there exists y ∈ R such that a+by is invertible in R. See [17] and [18]. In this note we will prove that strongly πregular rings have stable range one. As a consequence we shall obtain that modules satisfying Fitting’s Lemma (over any ring) cancel from direct sums. A ring R is said to be strongly π-regular if for each a ∈ R there exist a positive integer n and x ∈ R such that a = ax. By results of Azumaya [3] and Dischinger [8], the element x can be chosen to commute with a. In particular, this definition is left-right symmetric. Strongly π-regular rings were introduced by Kaplansky [12] as a common generalization of algebraic algebras and artinian rings. In [13], Menal proved that a strongly π-regular ring whose primitive factor rings are artinian has stable range one. In [11], various results concerning algebraic algebras and strongly π-regular rings were obtained. In particular, Goodearl and Menal showed that algebraic algebras over an infinite field have stable range one [11, Theorem 3.1] (in fact they showed the somewhat stronger condition called unit 1-stable range), and, in [11, p.271], they conjectured that any algebraic algebra has stable range one. Our Corollary 5 proves this conjecture. Further, they ask whether all strongly π-regular rings have stable range one [11, p.279], proving that the answer is affirmative in several cases. For instance, the strongly π-regular ring Received by the editors April 28, 1995. 1991 Mathematics Subject Classification. Primary 16E50, 16U50, 16E20.

K0 of purely infinite simple regular rings

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

Diagonalization of matrices over regular rings

R

Leavitt path algebras of separated graphs

is a purely infinite simple ring, then

Cancellation of projective modules over regular rings with comparability

K_0(R)^+= K_0(R)

STABLE RANK OF LEAVITT PATH ALGEBRAS

, the monoid of isomorphism classes of finitely generated projective

The exchange property for purely infinite simple rings

R

Morita Equivalence for Rings with Involution

-modules is isomorphic to the monoid obtained from

Multipliers of von neumann regular rings

K_0(R)