Witt rings of LG-rings
An LG-ring \(A\) is a commutative ring with \(1\neq 0\) such that the following local-global principle holds: if a polynomial \(f\in A[X_1,\cdots ,X_n]\), \(n\geq 1\), represents an invertible element over the localization \(A_{\mathcal P}\) for each prime ideal \({\mathcal P}\) of \(A\), then \(f\) represents an invertible element over \(A\). Let \(W(A)\) (resp. \(W_q(A)\)) be the Witt ring of regular bilinear (resp. regular quadratic) forms over the LG-ring \(A\). It is shown that there exists a bijection between the minimal prime ideals of \(W(A)\) and the signatures of \(A\) (resp. the set of maximal orderings of \(A\) provided \(| A/{\mathcal M}| \geq 3\) for each maximal ideal of \(A\)). Furthermore, the authors prove that if \(B\) is a Galois ring extension of \(A\) whose Galois group \(G\) is of odd order, then \(W(A)=W(B)^G\) (resp. \(W_q(A)=W_q(B)^G\)) with a suitably defined action of \(G\) on \(W(B)\) (resp. \(W_q(B)\)). Some of these results have been known before in the special case of semilocal rings (see, e.g., the lecture notes of \textit{R. Baeza} [Quadratic forms over semilocal rings, Lect. Notes Math. 655 (Berlin: Springer-Verlag, 1978; Zbl 0382.10014)]), and some of the arguments used in the case of semilocal rings carry over to LG-rings without major changes.
Witt ring of quadratic forms over a ring
scientific article; zbMATH DE number 1251263
scientific article