Question

Show that Z,+,× is a ring?

Answer 1

To show that Z,+,× is a ring, we need to prove properties such as additive closure, additive identity, and distributive property. By verifying these properties, we can conclude that Z,+,× is indeed a ring.

Step-by-step explanation:

A ring is a mathematical structure consisting of a set with two binary operations, usually denoted as + and ×. In order to show that Z,+,× is a ring, we need to prove the following properties:

1. Additive closure: For any two integers, their sum is also an integer.
2. Additive identity: There exists an element 0 such that for any integer a, a + 0 = 0 + a = a.
3. Additive inverse: For every integer a, there exists an element -a such that a + (-a) = (-a) + a = 0.
4. Multiplicative closure: For any two integers, their product is also an integer.
5. Multiplicative identity: There exists an element 1 such that for any integer a, a × 1 = 1 × a = a.
6. Distributive property: For any three integers a, b, and c, a × (b + c) = (a × b) + (a × c).

By verifying these properties, we can conclude that Z,+,× is a ring.