I'm looking for a specific construction, taking an abelian group (with designated element) $(G,+,1)$ to a commutative ring $(R,+,\cdot,1)$, where $G\subset R$ as a pointed abelian group, and which is universal in the following sense: for any commutative ring $(S,+,\cdot,1)$ and any map $f:G\rightarrow S$ preserving $+$ and $1$, there is an extension $g:R\rightarrow S$.

The idea was to give a general construction of rings from pointed abelian groups, which in particular constructs $(\mathbb Z, +, \cdot, 1)$ from $(\mathbb Z, +, 1)$. So, while it may add new elements in some cases, it is not adding more than necessary.

The reason for adding the requirement that the groups be pointed is that there are conceivably many choices for $1$ (especially in, say, $(\mathbb Q, +)$, where every nonzero element is equally suitable).