W. D. Burgess

On strongly πregular rings and homomorphisms into them

A ring S is strongly π-regular if for every a ∈ S there exists n ≥ 1 such that an∈ an+1S. It is first shown that the dominion and the maximal epimorphic extension of any ring homomorphism α:R →,S, S strongly π-regular, are strongly π-regular. Several results of Schofield on perfect and semiprimary rings are special cases. As an application it is shown that a strongly π-regular ring is a Schur ring, generalizing Lenagan’s theorem for artinian rings. Further if S ⊆ R where S is commutative strongly π-regular then for any finitely generated MSM⊗SR = 0, implies M = 0 , although ⊗Sis not faithful on homomorphisms. Another application shows that the dominion of the natural map Z → S acts like a characteristic for a strongly π-regular ring S A right noetherian ring R satisfying generic regularity, as defined by Goodearl, is shown to have a representation in a strongly π-regular ring [Rcirc] (which is also regular and biregular with simple images artinian) such that Spec [Rcirc] = Spec R , as sets, and Spec [Rcir...

On Embedding Rings in Clean Rings

A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that every ring can be embedded in a clean ring as an essential ring extension. It is seen that the centre of a clean ring need not be a clean ring. There is no “clean hull” of a ring. A family of examples is given where there is a ring R, not a clean ring, embedded in a commutative clean ring S so that there is no clean ring T, R ⊆ T ⊆ S, minimal with that property. It is also shown that a commutative pm ring (each prime ideal is contained in a unique maximal ideal) cannot be extended to a clean ring by the adjunction of finitely many central idempotents.

Homological Aspects of Semigroup Gradings on Rings and Algebras

This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup�. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the�-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert�-series in the associated path incidence ring. Therationalityofthe�-Eulercharacteristic, theHilbert�-seriesandthePoincar´ e-Betti�-seriesisstudied whenis torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

CLEAN CLASSICAL RINGS OF QUOTIENTS OF COMMUTATIVE RINGS, WITH APPLICATIONS TO C(X)

Commutative clean rings and related rings have received much recent attention. A ring R is clean if each r ∈ R can be written r = u + e, where u is a unit and e an idempotent. This article deals mostly with the question: When is the classical ring of quotients of a commutative ring clean? After some general results, the article focuses on C(X) to characterize spaces X when Qcl(X) is clean. Such spaces include cozero complemented, strongly 0-dimensional and more spaces. Along the way, other extensions of rings are studied: directed limits and extensions by idempotents.

THE REGULARITY DEGREE AND EPIMORPHISMS IN THE CATEGORY OF COMMUTATIVE RINGS

Elements of the universal (von Neumann) regular ring T(R) of a commutative semiprime ring R can be expressed as a sum of products of elements of R and quasi-inverses of elements of R. The maximum number of terms required is called the regularity degree, an invariant for R measuring how R sits in T(R). It is bounded below by 1 plus the Krull dimension of R. For rings with finitely many primes and integral extensions of noetherian rings of dimension 1, this number is precisely the regularity degree. For each n ≥ 1, one can find a ring of regularity degree n + 1. This shows that an infinite product of epimorphisms in the category of commutative rings need not be an epimorphism. Finite upper bounds for the regularity degree are found for noetherian rings R of finite dimension using the Wiegand dimension theory for Patch R. These bounds apply to integral extensions of such rings as well.

Elements of Minimal Prime Ideals in General Rings

Let R be any ring; a ∈ R is called a weak zero-divisor if there are r, s ∈ R with ras = 0 and rs = 0. It is shown that, in any ring R, the elements of a minimal prime ideal are weak zero-divisors. Examples show that a minimal prime ideal may have elements which are neither left nor right zero-divisors. However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors. The union of the minimal prime ideals is studied in 2-primal rings and the union of the minimal strongly prime ideals (in the sense of Rowen) in NI-rings.

The reduced ring order and lower semi-lattices

Every reduced ring

On the Quasi-Stratified Algebras of Liu and Paquette

R

On free commutative regular rings

has a natural partial order defined by

An analogue of the pierce sheaf for non-commutative rings

a\le b