free A-Algebra

Let \(X\) be a set and \(R\) a ring Then the free algebra \(\mathrm{freeAlg}_R(X)\) over \(X\) is defined as

\begin{align*} \mathrm{freeAlg}(X) = \bigoplus_{w \in X^{*}} Rw \end{align*}

where \(X^{*}\) is the kleene star (or free monoid) and \(\bigoplus\) the categorical coproduct (resp. ???) of R-Algebras