Proving ℤ[√2] is a Ring

Intro to Proofs · Axiom Academy

EXAMPLE Proving ℤ/2ℤ is a Ring Verify every ring axiom for the two-element set ℤ/2ℤ = 0, 1 under addition and multiplication mod 2. Let with addition and multiplication taken modulo 2 (so we keep only the remainder after dividing by 2 ). Show that is a commutative ring with unity by checking each ring axiom directly. Add, then keep the remainder on division by 2. Notice the single surprise: 1 + 1 = 0. Multiply, then keep the remainder on division by 2. Every product already lies in 0, 1 . Nicely done — you verified all of the ring axioms for , the smallest nontrivial ring. Here's what made it work: A finite ring is decided by its tables: with only two elements, the addition and multiplication tables list every possible product, so each axiom is a finite check you can read off directly. The signature relation is 1 + 1 = 0 : this is the one place arithmetic "wraps around." It makes 1 its own additive inverse ( -1 = 1 ) and gives its characteristic 2 . Identities sit on the diagonal: 0 leaves elements unchanged under + , and 1 leaves them unchanged under — visible as the unaltered header row/column of each table. Structural properties come for free: associativity, commutativity, and distributivity are inherited from the integers, because reducing mod 2 is a ring homomorphism . It's actually a field: the only nonzero element, 1 , satisfies , so it has a multiplicative inverse. is the field — the foundation of binary arithmetic and coding theory.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.