@EmilJeřábek Yes $x+y=a, x-y=b$ mplies equality

I wonder if it is necessary to consider "G is a subgroup of $Aut(R)$? I mean that according to the definition of action, it is implicity contained in the definition that $x\mapsto g.x$ is a Ring automorphism. On the other hand I think the first version of your answer satisfies the Ring automorphism property: the conjugation is a ring automorphism on the trancated polynomial ring $\frac{Z[x]}{x^2}$. Am I mistaken?

@GabeK Yes I see, thank you

i know that $S^2\times S^2$ is covering of itself so impossible to be covered by the three space you mentioned so I think I got my complete answer thanks for that. The only remaining point: Was the reference Griffith &Harris?

I know (without proof) the similar real case the covering by spher, plane and hyperbolic space.

@GabeK I guess the reference is Griffith& Harris, yes?

Thank you for your answer. I need time to understand it. BTW by cover you mean universal cover? What reference for the first paragraph?