Let $R$ denote a commutative ring with unity and $X$ a set of formal indeterminates. Then intuitively, the set of all formal polynomials in $X$ with coefficients in $R$ can be defined as the free commutative ring with unity generated by $R \cup X$ subject to the identities satisfied by $R$.

**Question 1.** Does this definition actually work?

If so, then I think that formal polynomials make sense in arbitrary algebraic structures. So my second question is:

**Question 2.** Irrespective of whether the above definition makes sense, do formal polynomials make sense in arbitrary algebraic structures?