Theorem·T23
The integers form a commutative ring with unity
Integer addition and multiplication satisfy all the ring axioms, multiplication is commutative, and 1 is a multiplicative identity.
In words
Integer multiplication is associative, commutative, and distributes over addition, and the integer 1 leaves every integer unchanged when multiplied: together with the additive group structure, the integers form a commutative ring with unity.
Never needed: F05 · F10 · F13 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1is an abelian group by T22.
- 2Associativity (R1). Let . Expanding both sides via two applications of D060 reduces (R1) to the natural-number identities each side of (R1) expanding, by distributivity and associativity of and commutativity/associativity of on , to the very same sum on both sides.
- 3Commutativity. and are equal since , , , by commutativity of multiplication, and then by commutativity of addition.
- 4Distributivity (R2). Let as above, so (D058). Expanding via D060 and D058: Each coordinate agrees by distributivity of over and commutativity/associativity of on : both first coordinates equal , both second coordinates equal . The other distributive law follows symmetrically.
- 5Unity. Write with . For : by D060. L11 gives and , and also , ; then and by commutativity and D027. So using D027. By commutativity, too.
- 6All clauses of D061 hold, is commutative, and is a unity: is a commutative ring with unity.
∎
Remarks
The first substantial ring in this library. Elementary ring facts now apply: e.g.
for every integer
. The ring structure, together with the total order, is what will eventually make
an ordered integral domain; that a nonzero product of nonzero integers is nonzero (no zero divisors) is not proved here but follows readily from the analogous fact on ω.