Proof

We must prove that (R[X],+,0) is a commutative group, that (R[X],*,1) is a monoid and that distributivity holds.

Since most verifications are very similar, we restrict to one typical verification, that of left distributivity.

Let a = a₀ + a₁X + ··· + a_nXⁿ, b = b₀ + b₁X + ··· + b_nXⁿ and c = c₀ + c₁X + ··· + c_nXⁿ be three polynomials. The coefficient of X^k in a * (b + c) equals

a₀(b_k + c_k) + ··· + a_k(b₀ + c₀)

and can be rewritten by commutativity and distributivity in R as

(a₀b_k + ··· + a_kb₀) + (a₀c_k + ··· + a_kc₀)

which is the coefficient of X^k in ab + ac.