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 c alled the regularity degree, an invariant for R measuring how R sits in T(R ). It is bounded below by I plus the Krull dimension of R. For rings with f initely many primes and integral extensions of noetherian rings of dimensio n 1, this number is precisely the regularity degree. For each n greater than or equal to 1, one can find a ring of regularity de gree n + 1. This shows that an infinite product of epimorphisms in the cate gory of commutative rings need not be an epimorphism. Finite upper bounds for the regularity degree are found for noetherian ring s R of finite dimension using the Wiegand dimension theory for Patch R. The se bounds apply to integral extensions of such rings as well.