Spectral calculations in rings

We examine the validity of certain spectral integral formulas in topological rings. We consider the sign and square-root functions in polymetric rings containing $\frac12$. It turns out that formal analogues of classical transformation kernels and the resolvent identity can be used to understand the situation. In the lack of $\frac12$, the functions $\frac12-\frac12\sgn(\frac12-z)$ and $\frac12-\sqrt{\frac14-z}$ can be generalized, respectively.