I wonder whether the quotient ring $R/I$ is always Dedekind-finite for any ring $R$ if $I$ is the two-sided ideal of $R$ generated by all elements of the form $xy-1$ where $yx=1$.
One might think that the answer is "obviously yes", but here's the problem: Forcing all one-sided units in $R$ to become two-sided units might introduce new one-sided (or even two-sided) units that do not arise as the coset of a one-sided (or two-sided) unit in $R$! For example, consider the quotient of the free ring on four generators $x_1$, $x_2$, $x_3$, and $x_4$ by the two-sided ideal generated by $x_1x_2-1$ and $x_3x_4-x_2x_1$. In this ring, $x_3$ is not a left inverse of $x_4$, but it will become a left inverse after one identifies $x_2x_1$ with $1$.