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This question seems to be so easy, but it will be helpful to be clear especially for students. Some define $x$ as asequence like $(0,1_{R},0...)$ which $R$ is a ring and other regard $x$ as a variable or an indeterminate. Alghough these denoting is absolutely applicable ones, which one is preferable for defining $x$?

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2 Answers 2

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The words "variable" and "indeterminate" apriori have no mathematical meaning. Therefore they cannot be used to define the polynomial ring over a certain ring. Moreover the word "variable" frequently causes the wrong impression that polynomials and polynomial maps are the same things, which is not the case in general.

A precise definition of the polynomial ring always requires to construct $x$ in one or the other way from a structure derived from $R$ itself.

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In algebra one often wants to have a structure with particular properties. For instance one likes the complex numbers to be a field that contains the real numbers, and also an element $i$ whose square is $-1$, and such that any complex number can be written in the form $a+bi$ with $a,b\in\mathbf{R}$. These properties are sufficient to reason about the complex numbers and develop all of its theory, but they are not sufficient to show that such a field exists. Indeed most people are initially sceptical about the fact that such a thing as a square root of $-1$ can exist anywhere. The definite proof that working with complex numbers will not lead to contradictions (absurdities), is to build a model of the complex numbers, which is well defined and in which all assumed properties hold. That model is (or better, can be) the set $\mathbf{R}^2$ of ordered pairs of real numbers, for which addition and multiplication are defined in a particular way, and in which $(0,1)$ plays the role of $i$. Once the existence of the structure of complex numbers has been proved beyond any doubt, it is however no longer necessary to use this model, and it is better to write $i$ instead of $(0,1)$ and more generally $a+bi$ instead of $(a,b)$. Indeed it would be very confusing to keep identifying $\mathbf{C}$ with $\mathbf{R}^2$, especially if one wants to consider vector spaces over $\mathbf{R}$ and over $\mathbf{C}$. Moreover there exist other models for $\mathbf{C}$ (for instance there is one using $2\times2$ matrices) and there is no reason to force using any particular model.

Similarly when considering polynmials, reasoning with an unknown (or variable) $x$ gives a good idea of the kind of manipulations and equalities (like $(x+1)(x-1)=x^2-1$) one wants to be valid. However, at some point one wants $x$ to not be an unknown number, but a valid value in itself; this is essential if one wants to talk about such things as degree (if $x$ were just a disguised number, then so would any polynomial in $x$, and numbers do not have a nonzero degree). Such a value is called an indeterminate, and I like to write it $X$ to stress it is grown-up and independent. But like for the imaginary unit $i$, one must justify the existence of a structure of polynomials, to be sure one doesn't risk contradictions when reasoning with them. To this end one can build a model for polynomials, which are infinite sequences of numbers (but ultimetely becoming all zero),for which addition and multiplication are defined in a particular way, and in which $(0,1,0,0,\ldots)$ plays the role of $X$. Once it has been established that one gets all the desired basic properties of polynomials in the model, the model can and should be forgotten.

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@Mark van Leeuwen: Thanks for complete hint. As you pointed,I also believe that what is established for $X$ or $x$ is EXACTLY the word "model". Thanks again. –  Babak S. Dec 6 '11 at 6:45

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