False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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FALSE: a square matrix over a commutative ring is invertible if and only if its determinant is nonzero

Statement

False claim: for every commutative ring $R$ and every positive-sized square matrix $A$ over $R$ , the matrix $A$ is invertible if and only if $\det (A) \neq 0$ .

Facts & Assumptions

Given: The claimed equivalence over arbitrary commutative rings.

[F1]

A ring in the published convention may be the zero ring (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

[F2]

$Z$ is a commutative ring (The integers form a commutative ring) and its only units are $1$ and $- 1$ ( $(Z, \cdot, 1)$ is a commutative monoid whose group of units is ${1, - 1}$ ; equivalently $u ∣ 1$ holds exactly for $u = 1$ and $u = - 1$ ).

[F3]

For a positive-sized square matrix $A = (a_{i j})$ over a commutative ring, $\det (A) = \sum_{σ \in S_{n}} sgn (σ) \prod_{i < n} a_{σ (i), i}$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L1]

The correct general criterion is: a positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit).

Refutation

technique · direct

1.1

The reverse implication fails in $Z$ . By [F3], the unique term in the $1 \times 1$ determinant gives $\det ([2]) = 2$ . This determinant is nonzero but is not a unit by [F2], so [L1] shows that $[2]$ is not invertible.

F2F3L1algebra

1.2

The forward implication also fails under the stated ring convention. In the zero ring allowed by [F1], the $1 \times 1$ identity matrix is $[0]$ ; it is its own two-sided inverse, and [F3] gives its determinant as $0$ .

F1F3algebra

2.1

Thus both directions of the claimed equivalence can fail. Replacing "nonzero" by "a unit" gives the valid equivalence [L1], including for the zero ring.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 100 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

András Pál, Introduction to Commutative Algebra, Proposition 7.2 (standard reference, not scraped)