Adjoining an inverse to a central element of an algebra Given a (not necessarily commutative) algebra $A$, and a central element $c \in A$, is it always possible to enlarge $A$ to a an algebra $A'$ in which $c$ is invertible? I guess one can take a set of generators and relations for $A$, and add a formal inverse $c^{-1}$ along with extra relations making $c^{-1}$ central and satisfying $c^{-1}c = c^{-1}c = 1$. But I guess this is a specific example of a general formal process. What could this process be?

Let $S=\{a^n \mid n \in \Bbb N\}$.
Then $S$ satisfies the Ore condition, so one can form the Ore localization $AS^{-1}$. This is constructed in the same way that localizations of commutative rings work. This is treated in many non-commutative algebra books, e.g. in Lam's "Lectures on Modules and Rings".
In this special case, there's another contruction yielding the same result: we can form the polynomial ring $A[x]$ and divide by the two-sided ideal $(ax-1)A[x]=A[x](ax-1)=A[x](ax-1)A[x]$ to get $A[x]/(ax-1)A[x]$. The class of $x$ will be a formal inverse of $a$.
