CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Over $Z / 4$ , a nonconstant polynomial can be a unit and product degree can drop

Statement refuted

For every commutative ring $R$ , every unit of $R [x]$ is constant and degrees add under multiplication of nonzero polynomials.

Facts & Assumptions

Given: The ring $R = Z / 4$ and the polynomial $u = 1 + 2 x \in R [x]$ .

[L1]

Over a commutative ring, product degree is at most the sum of the degrees, and the coefficient at that sum is the product of the leading coefficients (Degree inequalities for sums and products over a commutative ring).

[L2]

The ring $Z / 4$ is the quotient ring $Z / 4 Z$ , so $4 = 0$ in it (For every $n \in N$ , the congruence-class ring $Z / n$ is the quotient ring $Z / n Z$ ).

Counterexample

technique · direct

1.1

In $R [x]$ , $u^{2} = (1 + 2 x)^{2} = 1 + 4 x + 4 x^{2} = 1$ by [L2], so the nonconstant polynomial $u$ is a unit and is its own inverse.

givenL2algebra

2.1

Both factors $u$ have degree $1$ , but their product has degree $0$ ; [L1] permits this drop because the top coefficient is $2 \cdot 2 = 0$ in $R$ , so both parts of the statement are refuted.

step 1.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Neil Donaldson, Math 120B Notes, Section 22, coefficient-ring caveats (standard reference, not scraped)

Over Z/4Z/4, a nonconstant polynomial can be a unit and product degree can drop