THE REGULARITY DEGREE AND EPIMORPHISMS IN THE CATEGORY OF COMMUTATIVE RINGS

Abstract

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.

Keywords:

• AMS Classification: 13A99, 13B25, 13E15

ACKNOWLEDGMENT

This work was partially supported by grants A7539 and A7753 of the NSERC.

Notes

¹The referee has pointed out that the proofs in this paper can equally well be done without reference to sheaves.