On valuation rings that contain zero divisors

Let R be a commutative ring with identity. A new proof is given of the theorem due to Samuel and Griffin which states that R is integrally closed in its total quotient ring if and only if R is the intersection of B-valuation rings. We then prove the main result of the paper: If K is a r-regular ring, then K admits only Pruifer rings as valuation rings. 1. A ring means a commutative ring with identity. If (G, +) is a totally ordered abelian group, then a function v from a ring T into GU{oo} is an evaluation with value group G, if for all x, y E T: (1) v(xy)=v(x)+v(y); (2) v(z+y)>min{v(x), v(y)}; (3) v(1)=0 and v(0)= oo. If v maps Tonto GUJ{oo}, then v is called a valuation. For any value group G, let G+={cx E G:c >O}u{co}. Let R be a subring of the ring T and suppose that v is an evaluation with value group G. We consider four conditions on R. (V1) R=v-1(G+). (V2) v is a valuation and R=v1(G+). (V3) If P(X1, * , X,) is a dominated polynomial over R, then P(sj, , sr)?O for every set {sj}=j E T-R. A dominatedpolynomial over R is a polynomial of the form P(X1,. *, X,)=X'1( . r + 2 (6(l), *, (r)) where the order is given by the ordered product of r copies of the natural numbers. (V4} R contains a prime ideal P such that if B is a ring between R and T, and if Q is a prime ideal of B contracting to P, then R=B. If we assume that T is a field, then the above definitions are equivalent and correspond to the classical definition of a valuation domain. In [13] Received by the editors December 7, 1972. AMS (MOS) subject classifications (1970). Primary 13A15, 13B20, 13F05.