Polynomial ring over a finite commutative ring: addition, convolution and degree checks in (ℤ/4ℤ)[x]

Goal and scope

Over R=ℤ/4ℤ take f=1+2x and g=2x. Using the coefficient-sequence definition, compute and normalize f+g, f·g, f·f and g·g; prove the top-coefficient formula (fg)_{m+n}=f_m g_n and explain why deg(fg)=deg f+deg g fails here; finally prove that R[x] is not a finite ring.

Works inside the MKO's scope: the coefficient ring ℤ/4ℤ is given by its complete addition and multiplication tables (modulo 4), and polynomials are finite-support coefficient sequences normalized by removing trailing zeros (assumption a1); the coefficient ring is not assumed to be a field, integral domain, UFD or Euclidean domain, and the degree-additivity formula is not used in the presence of zero divisors (assumption a2). It does not treat polynomial ideals, quotient rings or finite fields, and does not change the canonical MKO or its English edition.

Givens and symbols

• finite_sum_rules
• finite_counting_basics
• mul_zero
• power_laws
• function_def

• G1 — R=ℤ/4ℤ={0,1,2,3}, with addition and multiplication given by the integer operation followed by the remainder on division by 4; this fixes both complete tables, and R is a finite commutative unital ring but not an integral domain (2·2=0).
• G2 — An element of R[x] is a coefficient sequence (a_k)_{k≥0} in R with only finitely many nonzero terms; as a list it is written with trailing zeros removed (normalization). Two polynomials are equal exactly when every coefficient agrees.
• G3 — f=1+2x, i.e. f_0=1, f_1=2 and all other coefficients 0; g=2x, i.e. g_0=0, g_1=2 and all others 0.

• R=\mathbb{Z}/4\mathbb{Z} — the coefficient ring, elements 0, 1, 2, 3
• R[x] — the polynomial ring over R
• f_k — the k-th coefficient of f (the coefficient of x^k)
• \deg f — for nonzero f, the largest k with f_k≠0
• e_n — the coefficient sequence of x^n: 1 at index n, 0 elsewhere

Rules and sources

• finite_sum_rules · prerequisite — Finite sums: Σ_{i=1}^{0} a_i=0 and Σ_{i=1}^{n+1} a_i=Σ_{i=1}^{n} a_i + a_{n+1}; Σ(a_i+b_i)=Σa_i+Σb_i; Σ c·a_i=c·Σa_i.
• finite_counting_basics · prerequisite — Finite counting: for disjoint finite sets A, B, |A∪B|=|A|+|B|; |A×B|=|A|·|B|; there is a bijection between finite sets ⇔ they have the same number of elements.
• mul_zero · prerequisite — Multiplying by zero: u·0=0. Reason: u·0=u·(0+0)=u·0+u·0; subtracting u·0 from both sides gives 0=u·0.
• power_laws · prerequisite — Integer exponent laws (base nonzero, or exponents positive integers): a^m·a^n=a^{m+n}, (a^m)^n=a^{mn}, (ab)^n=a^n b^n, a^0=1.
• function_def · prerequisite — Function: f:A→B assigns to each a∈A exactly one f(a)∈B. Two functions are equal ⇔ they have the same domain and codomain and agree at every a.
• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• def_unfold · prerequisite — Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true.
• de_morgan_logic · prerequisite — De Morgan's laws for logic: ¬(P∧Q) ⇔ ¬P∨¬Q and ¬(P∨Q) ⇔ ¬P∧¬Q.
• add_zero · prerequisite — Additive identity: u+0=u.
• injective_surjective_def · prerequisite — Injective: f(a)=f(a′) ⇒ a=a′. Surjective: every b∈B has some a∈A with f(a)=b. Bijective: both; then there is a unique inverse f⁻¹ with f⁻¹∘f=id_A and f∘f⁻¹=id_B.
• contradiction · prerequisite — Proof by contradiction: if assuming ¬P leads to a contradiction (a statement and its negation both hold), then P holds.
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• mko_polyring_def · cited — 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. · mko-polynomial-ring-over-finite-commutative-ring
• mod_reduce · prerequisite — In ℤ/nℤ an integer m stands for its remainder r on division by n (0≤r<n); two integers stand for the same element exactly when n divides their difference; sums and products are the remainders of the integer sum and product.
• poly_ops · prerequisite — Polynomial operations: (f+g)_k=f_k+g_k and (fg)_k=Σ_{i+j=k} f_i g_j (a finite sum computed in R).
• deg_def · prerequisite — The degree deg f of a nonzero polynomial f is the largest k with f_k≠0; the zero polynomial has no degree.
• L_top_coeff · lemma — Over any commutative ring, if deg f=m and deg g=n, then (fg)_k=0 for k>m+n and (fg)_{m+n}=f_m g_n. Hence deg(fg)=m+n exactly when f_m g_n≠0; otherwise fg=0 or deg(fg)<m+n.

Main derivation

\mathbb{Z}/4\mathbb{Z}\ \text{tables}

2+2=0,\ 2\cdot2=0,\ 1\cdot2=2,\ 1\cdot1=1,\ 1\neq0

f=1+2x,\ g=2x

f=(1,2),\ g=(0,2),\ \deg f=\deg g=1

f+g

(1+0,\ 2+2)=(1,0)

f+g=(1,0)

f+g=(1)=1,\ \deg(f+g)=0

(fg)_k=\sum_{i+j=k}f_ig_j

fg=(0,2,0)

fg=(0,2,0)

fg=(0,2)=2x,\ \deg(fg)=1\neq 1+1

f\cdot f

(1\cdot1,\ 1\cdot2+2\cdot1,\ 2\cdot2)=(1,0,0)=1

g\cdot g

(0\cdot0,\ 0\cdot2+2\cdot0,\ 2\cdot2)=(0,0,0)=0

\deg f=m,\ \deg g=n

(fg)_k=0\ (k>m+n),\ (fg)_{m+n}=f_mg_n

h=1+x=(1,1),\ g=(0,2)

hg=(0,2,2)=2x+2x^2,\ \deg(hg)=2

x^n=e_n,\ m\neq n

e_m\neq e_n

n\mapsto x^n\ \text{injective}

R[x]\ \text{is not finite}

x^2,\ x^4\in R[x]

x^2\neq x^4,\ \text{but}\ t^2=t^4\ \text{for every}\ t\in R

\text{S04, S06 to S09, S12}

f+g=1,\ fg=2x,\ f^2=1,\ g^2=0;\ R[x]\ \text{infinite}

Substeps and lemmas

(fg)_0=f_0g_0

1\cdot0=0

(fg)_1=f_0g_1+f_1g_0

1\cdot2+2\cdot0=2

(fg)_2=f_0g_2+f_1g_1+f_2g_0

0+2\cdot2+0=0

k>m+n,\ i+j=k

f_ig_j=0,\ \text{so}\ (fg)_k=0

(fg)_{m+n}=\sum_{i+j=m+n}f_ig_j

(fg)_{m+n}=f_mg_n

Reverse audits

• S14 · sufficient_goal_reduction · S04, S06, S07, S08, S09, S12 — Goal G: the computations, the degree explanation and infiniteness stated in the goal. Sufficient condition H: S04, S06 to S09 and S12; H⇒G by conjunction. H is proved forwards from G1 to G3; G is never used as a premise, and degree additivity is never assumed.
• S13 · premise_audit · G2, function_def, mod_reduce — Equality of the functions rests on checking all 4 points of R (a complete check, since the domain is finite); inequality of the polynomials rests on comparing coefficients. The two conclusions do not conflict, because the notions of equality differ.
• S12 · premise_audit · S11, finite_counting_basics, contradiction — It needs N+1 distinct elements for every N, which the general argument of S11 provides; it does not come from checking finitely many values of N.
• S11 · premise_audit · S01, G2, mko_polyring_def — The key premise is 1≠0: in the zero ring 1=0, every x^n equals 0, and R[x] has a single element.
• S10 · premise_audit · S09, L_top_coeff — The conclusion rests on the explicit check h_1 g_1=2≠0, not on 'degrees usually add'.
• S09.2 · inverse_equivalence · poly_ops, mul_zero, add_zero — The removed terms are all 0, and adding or removing zeros does not change the sum, so the two expressions are equal both ways.
• S09.1 · premise_audit · deg_def, mul_zero, de_morgan_logic — Only the definition of degree and multiplication by zero are used; it is not claimed that coefficients with k≤m+n are nonzero.
• S09 · premise_audit · S02, L_top_coeff — The lemma needs only a commutative ring, not an integral domain; it says when degrees add, not that they always do.
• S08 · inverse_equivalence · poly_ops, mod_reduce, mul_zero — Each coefficient recomputes to 0; the zero polynomial and (0,0,0) are the same sequence.
• S07 · inverse_equivalence · poly_ops, mod_reduce — The three coefficients can be recomputed one by one; f·f and 1 agree term by term.
• S06 · inverse_equivalence · G2, deg_def — Normalization is reversible (put the trailing zero back); the degree is read from the normalized list.
• S05.3 · inverse_equivalence · poly_ops, mod_reduce, mul_zero — All three terms can be recomputed; in ℤ, 2·2=4≠0, but in ℤ/4ℤ it is 0.
• S05.2 · inverse_equivalence · poly_ops, mod_reduce — Both terms are listed with none missing or repeated; the result can be recomputed.
• S05.1 · inverse_equivalence · mul_zero — The identity u·0=0 holds both ways.
• S05 · inverse_equivalence · poly_ops, mul_zero, finite_sum_rules — Each coefficient is an evaluation of the same convolution formula and can be recomputed term by term; both sides are the same polynomial.
• S04 · inverse_equivalence · G2, deg_def — Putting the trailing zero back gives the same sequence, so the step is two-way; the degree is read only from the normalized list.
• S03 · inverse_equivalence · poly_ops, mod_reduce — Backward: (1,0)−g=(1−0, 0−2)=(1,2)=f (in ℤ/4ℤ, −2=2), which recovers the left side f+g.
• S02 · inverse_equivalence · G2, deg_def — The lists and the polynomials determine each other: (1,2) reads back as 1+2x and (0,2) as 2x.
• S01 · premise_audit · G1, mod_reduce — Each entry can be recomputed by reduction (N1, N3); the step only reads the tables and derives nothing beyond them.

Error controls and negative controls

A finite coefficient ring does not give a finite polynomial ring

The misconception identified in the MKO is calling R[x] a finite carrier too. There are only 4 choices for each coefficient, but the degree is unbounded; S11 and S12 prove that R[x] contains arbitrarily many distinct monomials. For a nonnegative integer d, the subset whose coefficients of every exponent at least d vanish, including the zero polynomial, has 4^d elements. For d=0 it contains only the zero polynomial; for d=1 it is the constant subring. Both cases are multiplicatively closed. For d≥2 it is not closed: x^{d−1} and x belong to it, but their product x^d does not.

Degree additivity need not hold with zero divisors

The second misconception identified in the MKO is to force deg(fg)=deg f+deg g. Over ℤ/4ℤ, (2x)(2x)=0 has no degree, (1+2x)(2x)=2x has degree 1, and (1+2x)^2=1 has degree 0. The correct statement is L_top_coeff: degrees add exactly when the product of the leading coefficients is nonzero (S09, S10).

Normalize before reading the degree

Coefficientwise addition can produce trailing zeros: f+g first comes out as (1,0). (1,0) and (1) are the same polynomial, of degree 0, not 1 (S04). Remove trailing zeros before comparing or reading a degree.

• R=ℤ/4ℤ has only 4 elements, so R[x] is a finite ring → reject — 1≠0 makes 1, x, x² and so on distinct (S11), so R[x] has arbitrarily many elements (S12).
• deg((2x)(2x))=1+1=2 → reject — (2x)(2x)=4x²=0 is the zero polynomial, which has no degree (S08).
• x² and x⁴ take the same value at every element of ℤ/4ℤ, so x²=x⁴ in R[x] → reject — Polynomial equality is coefficientwise; the coefficients at index 2 are 1≠0 (S13).
• 1+2x has degree 1, so it cannot be a unit of R[x] → reject — In ℤ/4ℤ, (1+2x)²=1+4x+4x²=1 (S07), so 1+2x is indeed a degree-one unit. The integral-domain premise belongs to the rejected claim's degree-additivity argument against positive-degree units, not to this mod-4 identity. The coefficient ring here has zero divisors, so that degree argument cannot be applied.

Result and limitations

In (ℤ/4ℤ)[x], f+g=1 (degree 0 after normalization), fg=2x (degree 1≠2), f·f=1 (f=1+2x is a unit) and g·g=0 (two nonzero polynomials multiply to 0). Over any commutative ring (fg)_{m+n}=f_m g_n, so degrees add exactly when the product of the leading coefficients is nonzero. Since 1≠0, the monomials 1, x, x² and so on are distinct, and R[x] is not a finite ring, although R has only 4 elements.

• L_top_coeff is proved for every commutative ring; the computations in S03 to S08 and S10 belong to ℤ/4ℤ and these particular polynomials. Polynomial ideals, quotient rings and polynomials over finite fields (the MKO's next step) are outside this guide.

Verification and source binding

9 declared checks for this guide; only the listed expressions are covered.

• source commit: b28a5edbd46279b1db2a3f3f48bb3f447e18899f
• adopted source SHA-256: 483fdc8d305e2bb342f7008783ce4641fe203322aebbac3fe3dbe5604f4c2e83
• original source SHA-256: 0bab65c7935d902ed18c55a3ac5fa3655cf979c044b1d2746c31e5cb7d687baf
• Lean: NOT_RUN · independent AI: NOT_MEASURED

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