The singularity category as a stable module category

We investigate the stabilization $\mathcal{S}$ of the module category over an artinian ring $\mathrm{\Lambda }$ by formally inverting the tensor endofunctor given by the bimodule of relative noncommutative differential $1$-forms. It turns out that $\mathcal{S}$ is a Frobenius abelian category, which is equivalent to the category of finitely presented modules over the zeroth component $L_0$ of the Leavitt ring $L$. It follows that $L_0$ is an FC ring in the sense of Damiano, which is usually not quasi-Frobenius. Moreover, the singularity category of $\mathrm{\Lambda }$ is triangle equivalent to the stable module category over $L_0$.