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use the definition for a ring to prove that Z7 is a ring under the operations + and x as defined as follows: [a]7+[b]7 = [a+b]7 and [a]7 x [b]7 = [a x b]7 1. state each step of your proof 2. provide written justification for each step. But First I must
use the six definitions of a ring to prove that z7 satisfies them. A ring is a set R equipped with two binary operations,1 here denoted by + and *, that have the following properties. 1. (a + b) + c = a + (b + c) for all a, b and c in R (addition is associative).
2. a + b = b + a for all a and b in R (addition is commutative). 3. There is an element 0 ∈ R such that 0 + x = x + 0 = x for all x ∈ R. 4. For each element x ∈ R, there is a unique element y ∈ R such that x + y = y + x = 0. (We denote y by −x.) 5. (a * b)
* c = a * (b * c) for all a, b, and c in R (multiplication is associative). 6. (The distributive law) a * (b + c) = a * b + a * c and (b + c)* a = b * a + c * a for all a, b, and c in R

Use the definition for an integral domain to prove that Z7 is an integral domain.
1) state each step of your proof
2) provide written justification for each step of proofUsing definition
An integral domain is commutative ring with unity that has no zero divisors. (Nicodemi 86)