On the level of a semilocal ring with involution (Q1110564)

Let A be a commutative ring with 1, having an involution J (written as ``bar''). Define the level of (A,J), written s(A,J), to be the smallest integer n such that there exist \(a_ j\in A\) with \(-1=a_ 1\bar a_ 1+ \cdot \cdot \cdot +a_ n\bar a_ n.\) (If no such n exists then s(A,J) is infinite.) Suppose that A is semilocal and that there exists an element \(\mu\in A\) with \(\mu +{\bar \mu}=1.\) Furthermore assume that for every maximal ideal \({\mathfrak m}\) of A the residue field \(k=A/{\mathfrak m}\) is not too small (specifically, \(| k| >2\) and if J(\({\mathfrak m})={\mathfrak m}\) then \(| k| >4)\). The author develops the theory of hermitian Pfister forms to prove that if s(A,J) is finite it must be a power of 2. The case when J is the identity is discussed by \textit{R. Baeza} [Quadratic forms over semilocal rings, Lect. Notes Math. 655 (1978; Zbl 0382.10014)].