Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set

Theorem
Let $$I_X=\left\{1,X,X^2,\ldots\right\}$$ be   the  free monoid on a   singleton $$\{X\}$$.

Let  $$\left({R, +,  \circ}\right)$$ be a   ring with unity with   additive identity   $$0_R$$ and  multiplicative identity  $$1_R$$.

For two polynomials $$f,g$$ in the indeterminate $$X$$ over $$R$$, define the sum $$f\oplus g$$ to be the map $$f\oplus g:I_X\to R:X^i\mapsto f(X^i)+g(X^i)$$. Define their product $$f\otimes g$$ to be the map $$f\otimes g:I_X\to R:X^i\mapsto c_i$$, where $$c_i=\sum_{j+k=i}f(X^j)\circ g(X^k)$$.

Write $$R[X]$$ for the set of all polynomials in $$X$$ over $$R$$.

Then $$(R[X],\oplus,\otimes)$$ is a ring.