TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-03 (gpt-5.6-sol-codex-subscription)

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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.

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The integers form a commutative ring

Statement

$(Z, +, \cdot, 0, 1)$ with the operations of Arithmetic on the integers is a commutative ring with multiplicative identity, in which every element has the additive inverse $- [(a, b)] = [(b, a)]$ .

Facts & Assumptions

Given: $Z$ with the operations of Arithmetic on the integers.

[L1]

Addition on $N$ is commutative and associative, with zero as a two-sided identity (Addition is commutative, Addition is associative, Left identity for addition).

[L2]

Multiplication on $N$ is commutative and associative, distributes over addition, and has the usual zero and identity laws (Multiplication is commutative, Multiplication is associative, Distributivity and the successor law for multiplication, Zero and one under multiplication).

[L3]

The integer operations are independent of representatives (Integer addition and negation are well defined, Integer multiplication is well defined).

Proof

technique · direct

1.1L3suffices

By [L3], each axiom may be verified on arbitrary fixed representatives $(a, b), (c, d), (e, f)$ .

1.2L1

Associativity of $+$ : $((a, b) + (c, d)) + (e, f)$ and $(a, b) + ((c, d) + (e, f))$ both equal $(a + c + e, b + d + f)$ .

1.3L1

Commutativity of $+$ : $(a + c, b + d) = (c + a, d + b)$ .

1.4L1

Additive identity: $(a, b) + (0, 0) = (a + 0, b + 0) = (a, b)$ .

1.5L1

Additive inverses: $(a, b) + (b, a) = (a + b, b + a) \sim (0, 0)$ , since $(a + b) + 0 = (b + a) + 0$ .

1.6L1L2

Commutativity of $\cdot$ : swapping $(a, b) \leftrightarrow (c, d)$ sends $(a c + b d, a d + b c)$ to $(c a + d b, c b + d a)$ , the same pair.

1.7L1L2

Multiplicative identity: $(a, b) \cdot (1, 0) = (a \cdot 1 + b \cdot 0, a \cdot 0 + b \cdot 1) = (a, b)$ .

1.8L1L2

Associativity of $\cdot$ : expanding, both $((a, b) (c, d)) (e, f)$ and $(a, b) ((c, d) (e, f))$ equal $(a c e + a d f + b c f + b d e, a c f + a d e + b c e + b d f)$ .

1.9L1L2

Distributivity: $(a, b) \cdot ((c, d) + (e, f)) = (a (c + e) + b (d + f), a (d + f) + b (c + e)) = (a c + b d, a d + b c) + (a e + b f, a f + b e) = (a, b) (c, d) + (a, b) (e, f)$ .

2.1step 1.1step 1.2step 1.3step 1.4step 1.5step 1.6step 1.7step 1.8step 1.9∎

Steps 1.2–1.9 verify all axioms: $(Z, +, \cdot, 0, 1)$ is a commutative ring with identity and additive inverses.

The integers form a commutative ring

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources