Generalized Jacobian Conjectures -- A purely Algebraic Approach

Our goal is to settle a fading problem, the Jacobian Conjecture $(JC_n)$~: If $f_1, \cdots, f_n$ are elements in a polynomial ring $k[X_1, \cdots, X_n]$ over a field $k$ of characteristic zero such that $\det(\partial f_i/ \partial X_j)$ is a nonzero constant, then $k[f_1, \cdots, f_n] = k[X_1, \cdots, X_n]$. Practically, what we deal with is the generalized one, \noindent The Generalized Jacobian Conjecture$(GJC)$ :{\it Let $S \hookrightarrow T$ be an unramified homomorphism of Noetherian domains. Assume that $S$ is a simply connected UFD ({\sl i.e.,} ${\rm Spec}(S)$ is simply connected and $S$ is a unique factorization domain) and that $T^\times \cap S = S^\times$. Then $T = S$.} In addition, for consistency of the discussion, we raise some serious (or idiot) questions and some comments about the examples appeared in the papers published by the certain excellent mathematicians (though we are not willing to deal with them). However, the existence of such examples would be against our Main Result above, so that we have to dispute in Appendix B their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples which is shown explicitly.Comment: I do the previous manuscript cleaning and add three figures, and add Appendix