# Polynomial ring over a finite commutative ring

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite.

mko-polynomial-ring-over-finite-commutative-ring · definition · v0.10.0



## Mathematical statement

For a finite commutative unital ring R, each element of R[x] is a finite-support coefficient sequence. Addition is coefficientwise and multiplication is finite convolution. If 1 differs from 0, the monomials 1,x,x² and so on are distinct by degree, so ordinary R[x] is not a finite ring.





## Representative formula (exact TeX)

P(R)





## Assumptions


- The coefficient ring is specified by both complete finite commutative-ring tables; trailing zero coefficients are normalized.
- R is not assumed to be a field, integral domain, UFD or Euclidean domain; degree-of-product additivity is not used in the presence of zero divisors.





## Symbol scope


- P — polynomial-ring record
- R — finite coefficient ring





## Intuition, definition, and misconceptions

Trailing zeros do not change a polynomial and must be normalized.

Addition is coefficientwise and multiplication is finite convolution using operations in R.

A finite coefficient set does not bound the degree, so ordinary R[x] is generally infinite.

A polynomial ring over a finite coefficient ring need not have a finite carrier, and degree(fg)=degree(f)+degree(g) cannot be imposed unconditionally when zero divisors occur.





## Adopted derivation guide

[Polynomial ring over a finite commutative ring: addition, convolution and degree checks in (ℤ/4ℤ)[x]](https://ocme.evemisslab.com/en/derivations/derivation-mko-polynomial-ring-over-finite-commutative-ring-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite commutative ring](https://ocme.evemisslab.com/en/mko/mko-finite-commutative-ring/) — The parent ring tables supply coefficient addition and multiplication.
- [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/mko/mko-polynomial/) — Elements are represented by finite coefficient sequences.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Finite-support coefficients and convolution use indices.





## Verification boundary

Only explicit finite-support polynomial operations are calculated; no field, integral-domain, UFD, Euclidean-domain or finite-carrier claim is inferred.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add polynomial ideals, quotient rings and polynomials over finite fields.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/polynomial-ring-over-finite-commutative-ring.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-polynomial-ring-over-finite-commutative-ring/index.md)