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.

Question

embutiridsl · Accepted Answer

Step 1The requirements are a little vague, but it sounds like you&#039;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 &quot;representing the ring R[X] with integers.&quot;Further, the addition and multiplication are supposed to be &quot;based on&quot; the operations in Z, which I&#039;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&#039;s R or C, then you&#039;re never going to get an injective map of R[X] into Z, even as a set.Step 2Secondly, 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 &quot;let&#039;s just take some main features of addition and multiplication in Z and explore operations like that on other sets.&quot; Trying to cram rings back into Z is a bit of a step backwards

Alexia Avila · Accepted Answer

Step 1If 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 2For example, to add     m  ,  n  ∈      Z     we first unindex them to         i          −      1        (  m  )  ,        i          −      1        (  n  )  ∈  R  ,  , then perform the addition in R, then index the result, i.e. the transported addition     ⊕   in     S  =      Z     is  m  ⊕  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  ⊕  n  )  =      i          −      1        (  m  )  +      i          −      1        (  n  )  ,   and similarly for other operations, yielding a ring isomorphism     R  ≅      Z     with said transported operations.

Answered question

Answer & Explanation

New Questions in Algebra II