Polynomial ring over a finite commutative ring

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

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 — The parent ring tables supply coefficient addition and multiplication.
• Univariate polynomials over the rationals — Elements are represented by finite coefficient sequences.
• 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

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