Form of elements in the ring $R[X_0,X_1,\ldots]$

Question

I am sorry if my question is trivial.

Let $R$ be a commutative ring with unity. I understand that the elements of $R[X]$ ($X$ is undetermined) are just the polynomials over the ring $R$.

But what is the form of the elements in $R[X,Y]$ and as I asked in the title of this post, what is the form of elements in $R[X_0,X_1,\ldots]$.

Could you guys hint me to any good reference to understand more about these rings?

Answer 1

For elements in $R[X_1,\dots, X_n]$ one often uses the multi-index notation. This means we write $X = (X_1,\dots, X_n)$ and $X^i = X_1^{i_1}\dots X_n^{i_n}$, where $i = (i_1,\dots,i_n)$. Then an element of $R[X_1,\dots, X_n]$ is of the form $\sum_i a_i X^i$. That means you take finite sums of $X_1^{i_1}\cdots X_n^{i_n}$ together with coefficients in $R$. For example we have $2x^2 + xy + 43y^3 - 6z^5 \in \mathbb{R}[x,y,z]$. In the case of infinitely many indeterminates you still take finite sums that use finitely many of the variables.

Answer 2

$R[X,Y]$ is the ring of polynomials of two variables with coefficients in $R$. A general polynomial in two variables has the form $\sum_{i=0}^m\sum_{j=0}^n a_{ij}X^iY^j$ when $a_{ij}\in R$. In a very similar way you can define polynomials in more variables.

Now, if you have infinitely many variables $\{X_i\}_{i=0}^\infty$ (might be uncountable number as well) then $R[X_0,X_1,...]$ is the ring of all polynomials in these variables. But note that a polynomial is always a finite sum, so each element in $R[X_0,X_1,...]$ is actually a polynomial in finite number of variables from $\{X_i\}_{i=0}^\infty$. (the other coefficients are just zeros)

Answer 3

They're simply the polynomials in a countable set of indeterminates indexed by the set of natural numbers. By itself, each polynomial can be expressed with a finite set of indeterminates.

It also can be viewed as the direct limit of the inductive system $\bigl(R[X_0, \dots X_m]\hookrightarrow R[X_0, \dots X_n]\bigr)_{m\le n}$.

Example of a polynomial of total degree $3$ in two indeterminates, ordered by total degree of it monomials:

\begin{multline} f(X,Y)=a_0+ b_1X+c_1Y+a_2X^2+b_2XY+c_2Y^2\\+a_3X^3+b_3X^2y+c_3XY^2+d_3Y^3. \end{multline}