Let $R$ be a ring and $f\in R$. Is there something like an $f$-regular $K$-theory group of $R$ based on the category of $f$-regular $R$-modules, i.e. modules that do not have any $f$-torsion? If needed, I could assume that $R$ is an integral domain.
If $f$ is a non-zero-divisor in $R$ and $R$ is noetherian, then any finitely generated $R$-module has a resolution of length 1 by finitely generated modules upon which $f$ acts as a non-zero-divisor. One sees that by starting the resolution with a free module and then observing that a submodule of a free module has $f$ acting it as a non-zero-divisor. Now Quillen's resolution theorem applies to show that that the K-theory of the category of finitely generated $R$-modules is the same as the K-theory of the category of finitely generated $R$-modules on which $f$ acts as a non-zero-divisor.