A corollary of Krull's theorem says that in a commutative ring $R$ every non-unit lie in a maximal ideal of $R$. Well, my question is if this works for noncommutative rings.
So far, I got this: every non-left (right) invertible element of an arbitrary ring $R$ lie in a maximal left (right) ideal of $R$. Is this correct? Thanks in advance for your help.
That is correct. If $R$ is any ring with $1$ and $x \in R$ is not left-invertible, then $1 \notin Rx$, and therefore $Rx$ must be contained in a maximal left ideal. (You prove this in the same way as for commutative rings: The collection of left ideals containing $Rx$ but not $1$ is nonempty, so Zorn's lemma applies to show that it has a maximal element, which must be a maximal left ideal.)