TITLE: Formal Construction of Polynomial Ring in Several Variables
QUESTION [4 upvotes]: The formal construction of the polynomial ring in one variable is briefly the following:
We take a ring $(R,+,\cdot)$ with $1_R$.
We define $R^{ \mathbb{N}}$ be the set of all the sequences $(a_0,a_1,a_2,a_3,...)$, $a_i \in R^{ \mathbb{N}},\forall i\in \mathbb{N}$ and we define the following operations: 
$$+:  R^{ \mathbb{N}} \times  R^{ \mathbb{N}} \longrightarrow  R^{ \mathbb{N}},\  ((a_0,a_1,a_2,...),(b_0,b_1,b_2,...))\mapsto (a_0,a_1,a_2,...)+(b_0,b_1,b_2,...):= (a_0+b_0,a_1+b_1,a_2+b_2,...)$$ and $$\cdot:  R^{ \mathbb{N}} \times  R^{ \mathbb{N}} \longrightarrow  R^{ \mathbb{N}},\ ((a_0,a_1,a_2,...),(b_0,b_1,b_2,...))\mapsto (a_0,a_1,a_2,...) \cdot (b_0,b_1,b_2,...):=(c_0,c_1,c_2,...)$$
with $c_n=a_0b_n+a_1b_{n-1}+...+a_{n-1}b_1+a_nb_0,\forall n\in \mathbb{N}=\{0,1,...\}$.
Furthermore we define the equality $(a_0,a_1,a_2,...)=(b_0,b_1,b_2,...) \iff a_i=b_i, \forall i\in \mathbb{N}$. With these two binary operations we have that $(R^{ \mathbb{N}},+\cdot )$ is a ring with $1_{R^{ \mathbb{N}}}=(1_R,0_R,0_R,...)$. Now, polynomial is every element of the last ring of the form $(a_0,a_1,a_2,...,a_n,0_R,0_R,...)$. If $R[X]$ is the set of all the polynomials then $R[X]$ is a subring of $R^{ \mathbb{N}}$ and the mapping $f:R\longrightarrow R[X]$, $a\mapsto (a,0_R,0_R,...) $ is a monomorphism. So, we can say that $R$ is a subring of $R[X]$, and after this we have all the usual theorems.
My question is:
How we can do exactly the same construction in the polynomial ring in several variables with the same procedure?
PS 1: I apologize for my English. If you don't understand something ask me please.
PS 2: I know that a similar question already exists, but I think I have a different procedure.
Thank you in advance.

REPLY [0 votes]: Perhaps there is a way to do this as the "free-est" possible associative, unital $R$-algebra with unit (thus the algebra structures are defined by just morphisms of rings with unity from $R$ to the algebras) such that the set of variables commutes with the "copy" of $R$ and each other. That would define it even for an infinite set of variables $T$.
We could start with formal products in the set of variables, such that all but a finite number of exponents are $0$, where if all the exponents are $0$ we'll identify that with $1_{R[T]}$ then define the product of such unit coefficient monomials, then define formal multiplication with elements of $R$ to obtain the set of monomials and constants from $R$; then formal sums in the previous set such that all but a finite number of terms has $0_R$ as coefficient, where multiplication is extended linearly. Etc...
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