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WTThe integers modulo n form a commutative ring with unity

Theorem·T59

The residue classes modulo n form a commutative ring with unity, the class of 1.

For every , , with from D115 and from D116, is a ring; moreover is commutative and is a unity.
In words
The integers modulo n, with addition and multiplication of residue classes, form a ring in which multiplication is commutative and the class of 1 is a multiplicative identity.
Rests onF02F03F04F05F06F08F09F11F12F13F14A01A02A06A08
Never needed: F10 · A03 · A04 · A05 · A07 · A09 (computed from the citation graph, not asserted).

Proof

  1. 1
    Throughout, D115 and D116 give and (well defined by L74); each law below is inherited by pushing the corresponding law of (T23) through these formulas. Fix and write for .
  2. 2
    is an abelian group. Associativity: by associativity of on (T22). Commutativity: . Identity: . Inverse: , so is the additive inverse of (D059). Thus D038 holds and the group is abelian.
  3. 3
    Associativity of (R1). by associativity of on (T23, (R1)).
  4. 4
    Distributivity (R2). by distributivity on (T23, (R2)); the other law follows the same way.
  5. 5
    Commutativity and unity. ; and (T23).
  6. 6
    All clauses of D061 hold with the additive identity, is commutative, and is a unity: is a commutative ring with unity.
∎

Remarks

The class map is a surjective ring homomorphism : every axiom of is the image of the same axiom in . Unlike , this ring can have zero divisors: modulo , with neither factor zero. It is a field exactly when is prime, since then every nonzero class is invertible (L78, T61); for it degenerates to the trivial ring . This is the library's first quotient ring, and the setting for all of modular arithmetic.
Wikipedia ↗ZMod.commRing ↗

Used by

T61
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