Converting a polynomial ring to a numerical ring (transport of structure)

Question

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in parallel to ring of integers, I think converting a polynomial ring to a numerical (i.e. integer) ring is useful.

What I mean by converting to a numerical ring is: in the standard ring of integers, $+$ and $\cdot$ are defined as in usual arithmetic. But is there universal way of converting any polynomial/monomial rings such that each object in the ring gets converted to an integer, and $+$ and $\cdot$ can be defined differently from standard integer $+$ and $\cdot$? This definition would be based on integer arithmetic, though.

Answer 1

The requirements are a little vague, but it sounds like you'd like to find a function from a polynomial ring $R[X]$ into the set $\Bbb Z$, and then equip $\Bbb Z$ with a potentially unusual addition and multiplication to make the map a ring isomorphism, thereby "representing the ring $R[X]$ with integers."

Further, the addition and multiplication are supposed to be "based on" the operations in $\Bbb Z$, which I'm taking to mean that addition and multiplication between elements $a,b$ will somehow be a polynomial in the indeterminates $a,b$.

The first thing to notice is that the usefulness of this idea is immediately curtailed by the size of $R[X]$. If $R$ is uncountable, say it's $\Bbb R$ or $\Bbb C$, then you're never going to get an injective map of $R[X]$ into $\Bbb Z$, even as a set.

Secondly, one has to ask why it would be easier to work with $\Bbb Z$-with-a-bizzare-multiplication-and-addition rather than just $R[X]$. Take $\Bbb Z[X]$ for example: it would seem a lot simpler just to work in $\Bbb Z[X]$ directly.

Finally, the spirit of ring theory is "let's just take some main features of addition and multiplication in $\Bbb Z$ and explore operations like that on other sets." Trying to cram rings back into $\Bbb Z$ is a bit of a step backwards :)

Answer 2

If you accept subrings of the complex numbers as numerical rings, then just take a homomorphism $\mathbb Z[X] \to \mathbb C$ with nontrivial kernel. This reduces to sending $X$ to an algebraic number. A prime example is $X \mapsto i$, which gives you the numerical ring $\mathbb Z[i]$.

If you mean encoding the ring operations of $\mathbb Z[X]$ as integer operations, then this can be done by mapping $\sum_{k=0}^n a_k X^k$ to $\prod_{k=0}^n p_k^{a_k}$, where $p_k$ is the $k$-th prime. For other ideas along this line, see Gödel numbering for sequences.

Answer 3

If $\,R\,$ is a ring (or any algebraic structure) then one can transport the structure of $\,R\,$ to any set $\,S\,$ of the same cardinality, by push/pulling the algebra operations along any bijection of their underlying sets. When $\,S\,$ = $\,\Bbb N\,$ or $\,\Bbb Z,\,$ this can be viewed simply as indexing (or coding) the elements of $\,R\,$ (e.g. in computer representations of $\,R\,$ where indices are memory addresses).

For example, to add $\,m,n\in\Bbb Z\,$ we first unindex them to $\,i^{-1}(m),\,i^{-1}(n)\in R,\,$ then perform the addition in $\,R,\,$ then index the result, i.e. the transported addition $\,\oplus\,$ in $\,S=\Bbb Z\,$ is

$$ m \oplus n\, =\, i\,(i^{-1}(m)+i^{-1}(n))$$

and analogously for all other operations of $\,R.\,$ This implies that the unindex map $\,i^{-1}$ is a ring homomorphism, i.e. $\, i^{-1}(m\oplus n) = i^{-1}(m)+i^{-1}(n),\,$ and similarly for other operations, yielding a ring isomorphism $\, R\cong \Bbb Z\,$ with said transported operations.