Corollary 2

In a left Noetherian ring, every left or right invertible element is invertible.

Indeed, consider elements $x,y$ of a left Noetherian ring A such that $xy= 1$. Denote by $\boldsymbol{\delta }(x)$ and $\boldsymbol{\delta }(y)$, respectively, the endomorphisms $a\mapsto ax$ and $a\mapsto ay$ of the A-module $A_s$. We have $\boldsymbol{\delta }(y)\circ \boldsymbol{\delta }(x) = 1_{A_s}$; hence $\boldsymbol{\delta }(y)$ is surjective. By Corollary 1, a), $\boldsymbol{\delta }(y)$ is bijective. The endomorphism $\boldsymbol{\delta }(x)$ is then the inverse bijection of $\boldsymbol{\delta }(y)$, and we have $yx= (\boldsymbol{\delta }(x)\circ \boldsymbol{\delta }(y))(1) = 1$. The corollary follows.