The polynomials over a commutative ring $R$ are defined as usual: $R[x] = \{p \in R^\mathbb{N}: \#\mathrm{supp}(p) < \#\mathbb{N} \}$ where $\mathrm{supp}$ is the set of non-zero points and $\#$ is the cardinality of a set.
(2.13) Proposition: There is a unique commutative ring structure on the set of polynomials $R[X]$ having these properties:
(a) Addition of polynomials is vector addition.
(b) Multiplication of monomials is given by $ax^m * bx^n = > (ab)x^{m+n}$
(c) The ring $R$ is a subring of $R[X]$ when the elements of $R$ are identified with the constant polynomials.
The proof of this proposition is notationally unpleasant without having any interesting features, so we omit it. $\square$
What is so notationally unpleasant about this proof? It seems to me like we could simply do induction on degree of polynomials to show multiplication of them is the usual result, applying (a) and (b) at the induction basis. Then to show it's unique we could consider the identity map in $R[X]$ and show that it's a ring homomorphism. Am i missing something? Also this proposition is in the section before ring homomorphisms are introduced. Does the word "unique" have meaning without isomorphisms? Finally, the last condition seems superflous. Since constants are monomials, wouldn't (c) follow from (b)?
