Cyclotomic equations and square properties in rings (Q1075351)

Let R be a commutative ring with an identity (not a field). We say that R satisfies square property one if the ring R satisfies SP1a: If \(r\in R\) and \(x^ 2+x+1=0\) has solutions mod (r) then \(r=\pm (u^ 2+v^ 2+uv)\) for some ring elements u,v. SP1b: If \(r=\pm (u^ 2+v^ 2+uv)\) for some u,v\(\in R\) with \((u,v)=1\) then \(x^ 2+x+1=0\) has solutions mod (r). Here \((u,v)=1\) indicates that u and v have no common divisors. Using the structure of the projective special linear group PSL(2,R) the author shows that the following rings satisfy square property one. (a) \({\mathbb{Z}}\). (b) \({\mathbb{Z}}_{p^ n}\) where \(n>1\) and p is a prime such that -3 is not a square mod p. (c) F[x] where F is a field of characteristic \(\neq 2\), with -3 not a square in F and every matrix of trace 1 in PSL(2,F) is conjugate in PSL(2,F) to either \(\pm \left( \begin{matrix} 0\\ 1\end{matrix} \begin{matrix} -1\\ 1\end{matrix} \right)\) or \(\pm \left( \begin{matrix} -1\\ -1\end{matrix} \begin{matrix} 1\\ 0\end{matrix} \right)\). (d) Euclidean domains D of characteristic \(\neq 2\) with trivial units and a sub-additive norm function satisfying \(0\neq N(b)\leq N(a)\) implies \(N(a+kb)<N(a)\) for some \(k\in D.\) Finally he shows that the class of sum of squares rings (rings which satisfy Fermat's two square theorem) and the class of rings satisfying square property one are non-coincidental.