How to prove that if $x$ is not invertible, then $Rx \neq R$?

Question

Let $R$ be an associative ring with identity and let $x \in R$ be an element that is not invertible. How can we prove that $Rx \neq R$? Here $Rx$ denotes the left ideal generated by $x$.

Answer 1

As Robert Israel said, the correct statement is "if $x$ is not invertible, then $Rx \neq R$ or $xR \neq R$", and this is easy to prove by the contrapositive: if $Rx = R$ and $xR = R$, then there are $y, z \in R$ such that $xy = 1$ and $zx = 1$, and it is easy to conclude that $y = z$, hence $x$ is invertible.