Rings and Fields

Walkthrough

Verify that (ℤ, +, ·) is a ring.

1. 1.A ring (R, +, ·) must satisfy: (1) (R, +) is an abelian group, (2) · is associative, (3) distributive laws hold.
2. 2.(ℤ, +): addition is associative and commutative, the identity is 0, and every integer n has inverse −n. So (ℤ, +) is an abelian group. ✓
3. 3.Multiplication is associative: (a·b)·c = a·(b·c) for all integers. ✓
4. 4.Distributivity: a·(b + c) = a·b + a·c and (a + b)·c = a·c + b·c. ✓
5. 5.So (ℤ, +, ·) is a ring. (It is also a commutative ring with identity 1.)