Sums of powers in rings and the real holomorphy ring.

*(English)*Zbl 0922.12003The real holomorphy ring of a commutative ring \(R\) is defined to be the subring \(H(R)\) of elements which are globally finite on the real spectrum of \(R\). The authors present an extensive investigation of such real holomorphy rings. Suppose \(A\) is a real affine domain over \({\mathbb R}\), \(F\) is the quotient field of \(A\) and \(V\) is the associated real affine variety. It is proved that \(A\subseteq H(F)\) iff the set of all central real points of \(V\) is compact. Moreover, if the subspace of singular, real points of V is compact then \(H(A)=A\cap H(F)\). Considering the sequence \(H(R), H(H(R)), \ldots ,H^n(R)\) of iterated real homomorphism rings, the authors show that if \(\dim R=d\), then \(H^{d+1}(R)=H^d(R)\).

For the semigroup of sums of higher powers holds the following: \[ \text{Sat}(1+\sum R^2)=\text{Sat}(1+\sum R^{2n})= \bigcap_{{\mathbf p}\in\text{Spec } R, {\mathbf p}\text{-real}} (R\setminus {\mathbf p}), \] where \(\text{Sat}(S)=\{s \in R \mid \exists_{s'\in S} ss'\in S\}\) for any multiplicative semigroup \(S\) in \(R\).

The most complete results are obtained for rings satisfying \(1+\sum R^2 \subseteq R^*\). In this case, every subring of \(R\) containing \(H(R)\) is a Prüfer ring in \(R\). The connection between ideals and units of \(R\) is established by the following isomorphism of groups: \[ (R^*\cap \sum R^{2n})/(R^*)^{2n}\cdot (H(R)^* \cap \sum R^2) \cong \text{Cl}(H(R),R), \] where \(\text{Cl}(H,R)\) is the class group of invertible fractional ideals. The Waring \(n\)-constant is the smallest integer \(l\) such that every element in \(R^*\cap \sum R^n\) is a sum of \(l\) \(n\)-th powers of elements of \(R\), or \(\infty\) if such \(l\) does not exist. The authors prove that the Waring \(2n\)-constant for \(R\) is finite iff the Waring 2-constant is finite. Moreover, for a certain class of rings an effective upper bound for the Waring \(2n\)-constant is determined.

For the semigroup of sums of higher powers holds the following: \[ \text{Sat}(1+\sum R^2)=\text{Sat}(1+\sum R^{2n})= \bigcap_{{\mathbf p}\in\text{Spec } R, {\mathbf p}\text{-real}} (R\setminus {\mathbf p}), \] where \(\text{Sat}(S)=\{s \in R \mid \exists_{s'\in S} ss'\in S\}\) for any multiplicative semigroup \(S\) in \(R\).

The most complete results are obtained for rings satisfying \(1+\sum R^2 \subseteq R^*\). In this case, every subring of \(R\) containing \(H(R)\) is a Prüfer ring in \(R\). The connection between ideals and units of \(R\) is established by the following isomorphism of groups: \[ (R^*\cap \sum R^{2n})/(R^*)^{2n}\cdot (H(R)^* \cap \sum R^2) \cong \text{Cl}(H(R),R), \] where \(\text{Cl}(H,R)\) is the class group of invertible fractional ideals. The Waring \(n\)-constant is the smallest integer \(l\) such that every element in \(R^*\cap \sum R^n\) is a sum of \(l\) \(n\)-th powers of elements of \(R\), or \(\infty\) if such \(l\) does not exist. The authors prove that the Waring \(2n\)-constant for \(R\) is finite iff the Waring 2-constant is finite. Moreover, for a certain class of rings an effective upper bound for the Waring \(2n\)-constant is determined.

Reviewer: M.Kula (Katowice)