ASSUMPTIONS

A set, R, with two operations, + and *, is a ring if the following properties are shown to be true:

1. Closure property of addition: for all s and t in R, s + t is also in R

2. Closure property of multiplication: for all s and t in R, s * t is also in R

3. Additive identity property: there exists an element 0 in R such that s + 0 = 0 + s = s for all s in R

4. Additive inverse property: for every s in R, there exists t in R, such that s + t =t + s = 0

5. Associative property of addition: for every q, s, and t in R, q + (s + t) = (q + s) + t

6. Associative property of multiplication: for every q, s, and t in R, q * (s * t) = (q * s) * t

7. Commutative property of addition: for all s and t in R, s + t = t + s

8. Left distributive property of multiplication over addition: for every q, s, and t in R, q * (s + t) = q * s + q * t

9. Right distributive property of multiplication over addition: for every q, s, and t in R, (s + t) * q = s * q + t * q

Given the set of integers mod m denoted Z_m, the elements of Z_m are denoted [x]_m, where x is an integer from 0 to m- 1. Each element [x]_m is an equivalence class of integers that has the same integer remainder as x when divided by m.

The set of integers Z forms a ring with the usual operations of integer addition and multiplication. Given this fact, you are asked to prove that Z_m for an assigned value of m also has properties of a ring in part A of this task. Each step of each proof must be justified using an appropriate property from the ring Z or the given definitions of modular arithmetic operations.

A. Using the fact that Z is a ring, prove the following sixproperties of a ring also hold for the set Z_m. Use the properties of a ring in the Assumptions section and the given definitions of the operations of modular addition and multiplication. Do notuse Cayley tables or exhaustive lists.

m=56

1. Prove the closure property of addition for Z_m. Using the notation defined in the Assumptions section, justify each step, including naming the specific property or operation definition that applies to that step.

2. Prove that an additive identity element exists in Z_m and show that the additive identity property holds for Z_m. Using the notation defined in the Assumptions section, justify each step, including naming the specific property or operation definition that applies to that step.

3. Prove that additive inverse elements exist in Z_m and show that the additive inverse property holds for Z_m. Using the notation defined in the Assumptions section, justify each step, including naming the specific property or operation definition that applies to that step.

4. Prove the associative property of addition for Z_m. Using the notation defined in the Assumptions section, justify each step, including naming the specific property or operation definition that applies to that step.

5. Prove the commutative property of addition for Z_m. Using the notation defined in the Assumptions section, justify each step, including naming the specific property or operation definition that applies to that step.

6. Prove the left distributive property of multiplication over addition or the right distributive property of multiplication over addition for Z_m. Using the notation defined in the Assumptions section, justify each step, including naming the specific property or operation definition that applies to that step.