A new proof of a known special case of the Jacobian Conjecture

The famous Jacobian Conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ with invertible Jacobian, is invertible ($K$ is a characteristic zero field). A known result says that if $K[f(x),f(y)]\subseteq K[x,y]$ is an integral extension, then $f$ is invertible. We slightly generalize this known result to the following: If for some “good” $\lambda \in K$ (in a sense that will be explained) $mK[x,y]\mathrm{\ne }K[x,y]$ for every maximal ideal $m$ of $K[f(x),f(y)][x+\lambda y]$, then $f$ is invertible. We also apply our ideas to the Jacobian Conjecture, without any further assumptions.