Associative Local Function Rings

We prove that for an arbitrary field $k,$ a complete, associative $k^r$-algebra $\widehat{H}$ augmented over $k^r$ has exactly $r$ maximal two-sided ideals and deserves the name $r$-pointed. If $A$ is any $k$-algebra, $M=\{M_i{\}}_{i=1}^r$ is a family of simple right $A$-modules with a countable $k$-basis, and there is a homomorphism ${\rho }_A:A\to {\text{enm}}_{\widehat{H}}(H{\widehat{\otimes }}_{k^r}({\oplus }_{i=1}^r{M}_i))=:\widehat{O}(M)$ then $\widehat{O}(M)$ is $r$-pointed and $M$ is contained in the set of right simple $\widehat{O}(M)$-modules. Our main result is that the subalgebra generated ${\rho }_A(A)$ and all ${\rho }_A(a{)}^{-1}$ whenever ${\rho }_A(a)$ is a unit, is a natural substitute for the localization $A(M)$ of the $k$-algebra $A$ in $M$ which only exists when the Ore condition is fulfilled.