Celeste Barajas

2022-11-23

Converting a polynomial ring to a numerical ring (transport of structure)
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 ⋅ 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 ⋅ can be defined differently from standard integer + and ⋅? This definition would be based on integer arithmetic, though.

embutiridsl

Expert

Step 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 Z, and then equip 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 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 R or C, then you're never going to get an injective map of R[X] into Z, even as a set.
Step 2
Secondly, one has to ask why it would be easier to work with Z-with-a-bizzare-multiplication-and-addition rather than just R[X]. Take Z[X] for example: it would seem a lot simpler just to work in Z[X] directly.
Finally, the spirit of ring theory is "let's just take some main features of addition and multiplication in Z and explore operations like that on other sets." Trying to cram rings back into Z is a bit of a step backwards

Alexia Avila

Expert

Step 1
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 $S=N$ or 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).
Step 2
For example, to add $\phantom{\rule{thinmathspace}{0ex}}m,n\in \mathbb{Z}\phantom{\rule{thinmathspace}{0ex}}$ we first unindex them to $\phantom{\rule{thinmathspace}{0ex}}{i}^{-1}\left(m\right),\phantom{\rule{thinmathspace}{0ex}}{i}^{-1}\left(n\right)\in R,\phantom{\rule{thinmathspace}{0ex}}$, then perform the addition in R, then index the result, i.e. the transported addition $\phantom{\rule{thinmathspace}{0ex}}\oplus \phantom{\rule{thinmathspace}{0ex}}$ in $\phantom{\rule{thinmathspace}{0ex}}S=\mathbb{Z}\phantom{\rule{thinmathspace}{0ex}}$ is
$m\oplus n\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}\left({i}^{-1}\left(m\right)+{i}^{-1}\left(n\right)\right)$
and analogously for all other operations of R. This implies that the unindex map $\phantom{\rule{thinmathspace}{0ex}}{i}^{-1}$ is a ring homomorphism, i.e. $\phantom{\rule{thinmathspace}{0ex}}{i}^{-1}\left(m\oplus n\right)={i}^{-1}\left(m\right)+{i}^{-1}\left(n\right),\phantom{\rule{thinmathspace}{0ex}}$ and similarly for other operations, yielding a ring isomorphism $\phantom{\rule{thinmathspace}{0ex}}R\cong \mathbb{Z}\phantom{\rule{thinmathspace}{0ex}}$ with said transported operations.

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