New bad lines in R[x, y] and optimization of the Epimorphism Theorem

We give necessary and sufficient conditions on a commutative ring with unity R so that the following holds: any polynomial f in R[x, y] such that R[x, y]/(f) similar or equal to R-[1] (a line for short) is a variable (= alpha(x) for some automorphism alpha:R[x, y] -> R[x, y]). The main ingredient here is the construction of new bad lines (lines which are not variables). (c) 2006 Elsevier Inc. All rights reserved.