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In our course we introduced the concept of polynomials as a part of a more general construction, namely the monoid ring (or even more general, the monoid algebra) $R[M]$, where $(R,+_R,\cdot_R)$ is a ring and $(M,+_M)$ is a monoid. Then we introduced the notation $R[x]:=R[\mathbb{N}]$. So "$R[x]$" has a precisely defined meaning (Just like $R[\mathbb{Z}]$ or $R[\mathbb{Z}_{12}]$ would have).

So far so good. But now my problem pops up: Somewhat later in the course we suddenly have to deal with $F[u^2,t^3]$, $F[x,x^{-1}]$ etc. (where $F$ is a field)...but these haven't got any meaning associated. We treated $F[u^2,t^3]$, for example, as "polynomials for which we substitute all monomials $x,y$ with $u^2,t^3$". With this explanation I can operate with elements from $F[u^2,t^3]$, but I actually don't know what they are (in opposition to elements from $R[\mathbb{N}]$, which are function $\mathbb{N} \rightarrow R$ that are $0_R$ almost everywhere).

I tried to give some meaning to the above expressions $F[u^2,t^3]$, $F[x,x^{-1}]$ by trying to somehow apply a so-called "evaluation homomorphism" of which I only have a rather vague idea - I think it tells me, when I can use a polynomial as a "pattern" by which I can create other objects, by plugging the new object in place of the "indeterminates" - those would be in my case, I think, again polynomials. So I use a polynomial in which I put another polynomial; but I couldn't make this precise, because at this stage all the concepts are just to fuzzy for me to use them.

Could you please help me clarify what, for example, $F[x^2,x^3]$ really means ("set-theoretically" - just like I specified the elements of $ R[\mathbb{N}]$ above) and why I can work with them in the way described above?

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I think you're right: there are two slightly different notions. The first one is the adjunction of indeterminates to a given (commutative) ring. The second one is this: If $R$ is a ring, if $S$ is a subring, and if $a,b,\dots$ are elements of $R$, then $S[a,b,\dots]$ denotes the subring of $R$ generated by $S$ and $a,b,\dots$ –  Pierre-Yves Gaillard Nov 29 '11 at 16:49
    
From the monoid-ring perspective, the subring $F[u^2,t^2] \subset F[u,t]$ corresponds to the monomial submonoid $\:2\:\mathbb N\times 3\:\mathbb N\subset \mathbb N^2\:.$ So the polynomials,viewed as maps from monomials to coefficients, naturally restict to be viewed as maps from $\:2\:\mathbb N\times 3\:\mathbb N \to F\:.$ –  Bill Dubuque Nov 29 '11 at 17:03
    
You can define $M$ to be the submonoid of $\mathbb N \times \mathbb N$ of elements of the form $(2k,3l)$. Then $F[u^2,t^3]$ is just $F[M]$, and the inclusion $M\rightarrow \mathbb N \times \mathbb N$ yields the natural inclusion $F[M]\rightarrow F[u,t]$ –  Thomas Andrews Nov 29 '11 at 17:30
    
@ThomasAndrews Did you miss my above comment? Yours seems the same. –  Bill Dubuque Nov 29 '11 at 18:06
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3 Answers

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The most general meaning of $R[\alpha,\beta,\ldots]$ is "the smallest ring that contains $R$ as a subring and also contains the elements $\alpha$, $\beta$, et cetera".

Usually there will be a natural larger ring $R'$ that contains $R$ as subring and also contains $\alpha,\beta,\ldots$ as elements. In that case $R[\alpha,\beta,\ldots]$ is the intersection of all subrings of $R'$ that contain $\{R\}\cup\{\alpha,\beta,\ldots\}$. It is easy to see that this intersection is indeed a subring of $R'$.

  • In the example $F[u^2,t^3]$ it is natural to take $R'$ to be the polynomial ring $F[u][t]$. Then $F[u^2,t^3]$ is the subring of $F[u][t]$ of polynomials where the power of $u$ is always even and the power of $t$ is always a multiple of $3$.

  • For $F[x,x^{-1}]$ one could take $R'$ to be the ring of rational functions in $x$.

  • For $F[x^2,x^3]$ you can take $R'$ to be either the ring of rational functions, or just $F[x]$. Taking $R'$ to be too big usually does not change the answer, as long as it is unambiguous how arithmetic on the $\alpha$s and $\beta$s is supposed to work.

  • The polynomial ring $R[x]$ itself can also be understood as "the smallest ring that contains $R$ and $x$", where $x$ is implicitly understood not to obey any equations but those that follow from the ring axioms and the existing arithmetic in $R$. However, in this case the description does not really work as a construction because we have no a priori existing ring extension for $x$ to be a member of. One could use the ring of "quotients of all formal expressions built from "$x$" and a constant for each member of $R$, by the equivalence relation of all identities that can be proved from the ring axioms and all true statements $a+b=c$ and $ab=c$ with $a,b,c\in R$". But that is something of a mouthful -- the monoid ring is an easier construction.

A different but equivalent way of looking at it is through the evaluation homomorphism. Recall that $R[x,y]$ has the property that any homomorphism $\varphi:R[x,y]\to S$ is determined completely by its restriction to $R$ together with the values of $\varphi(x)$ and $\varphi(y)$. In particular if $R$ is a subring of $S$ and we specify that $\varphi$ must be the identity on $R$, then we only need the values of $\varphi(x)$ and $\varphi(y)$ to specify $\varphi$ completely.

Then $R[\alpha,\beta]$, where $\alpha,\beta\in S$ is just the subring of $S$ that is the image of $R[x,y]$ under the homomorphism that fixes $R$ and takes $x$ to $\alpha$ and $y$ to $\beta$.

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There are a couple different ways of thinking about it.

If you consider the additive submonoid $M:=\{0,2,3,\cdots\}=(\mathbb{N}\cup\{0\})\setminus \{1\}$ of $\mathbb{N}\cup\{0\}$, then $F[x^2,x^3]$ can be thought of as $F[M]$.

Alternatively, consider the homomorphism $f:F[x,y]\rightarrow F[x]$ where $f(x)=x^2$, $f(y)=x^3$, and $f(\alpha)=\alpha$ for all $\alpha\in F$ (ie $f$ is the evaluation homomorphism taking $x$ to $x^2$ and $y$ to $x^3$--if it isn't clear that $f$ is a ring homomorphism, it would be worth sitting down and proving it). Then $F[x^2,x^3]$ can be thought of as the image of $f$.

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For the purposes of this problem, $0\in \mathbb N$, or otherwise $\mathbb N$ would not be a monoid under addition. –  Thomas Andrews Nov 29 '11 at 17:41
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In general, if $M$ is a monoid, and $M'$ is a sub-monoid, then there is a natural inclusion of $R[M']$ in $R[M]$. Given the ring $F[u,t] = F[\mathbb N \times \mathbb N]$, $F[u^2,t^3]$ is the sub-ring corresponding to the monoid $M'=\{(2k,3l)| k,l\in \mathbb N\}$.

$F[x,x^{-1}]$ can be seen as $F[\mathbb Z]$.

$F[x^2,x^3]$ is just $F[M]$ where $M$ is the sub-monoid of $\mathbb N$ generated by $2$ and $3$ - that is, which is really just $\{n\in \mathbb N: n\neq 1\}$.

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