R. Raphael

The epimorphic hull of C(X)

Abstract The epimorphic hull H(A) of a commutative semiprime ring A is defined to be the smallest von Neumann regular ring of quotients of A . Let X denote a Tychonoff space. In this paper the structure of H(C(X)) is investigated, where C(X) denotes the ring of continuous real-valued functions with domain X . Spaces X that have a regular ring of quotients of the form C(Y) are characterized, and a “minimum” such Y is found. Necessary conditions for H(C(X)) to equal C(Y) for some Y are obtained.

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.

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.

Searching for Absolute

In previous papers, Barr and Raphael investigated the situation of a topological space Y and a subspace X such that the induced map C(Y) → C(X) is an epimorphism in the category CR of commutative rings (with units). We call such an embedding a CR-epic embedding and we say that X is absolute CR-epic if every embedding of X is CR-epic. We continue this investigation. Our most notable result shows that a Lindelspace X is absolute CR-epic if a countable intersection ofX- neighbourhoods of X is aX-neighbourhood of X. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfectmap. A strengthening of the Lindel¨ of property leads to a new class with the same closure properties that is also closed under finite products. Moreover,all �-compact spaces and all Lindel¨ of P-spaces satisfy this stronger condition. We get some results in the non-Lindelcase that are sufficient to show that the Dieudonn´ e plank and some closely related spaces are absolute CR-epic.

\mathcal{CR}

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.

-Epic Spaces

Every reduced ring

THE REGULARITY DEGREE AND EPIMORPHISMS IN THE CATEGORY OF COMMUTATIVE RINGS

R

The reduced ring order and lower semi-lattices

has a natural partial order defined by

ON PRODUCTIVELY LINDELOF SPACES

a\le b

A minimal regular ring extension of C(X)

if