**Given: A set R with two operations + and * is a ring if the following 8 properties are shown to be true:**

**1. closure property of +: For all s and t in R, s+t is also in R**

**2. closure property of *: For all s and t in R, s*t is also in R**

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Order Paper Now**3. additive identity property: There exists an element 0 in R such that s+0=s for all s in R**

**4. additive inverse property: For every s in R, there exists t in R, such that s+t=0**

**5. associative property of +: For every q, s, and t in R, q+(s+t)=(q+s)+t**

**6. associative property of *: For every q, s, and t in R, q*(s*t)=(q*s)*t**

**7. commutative property of +: For all s and t in R, s+t =t+s**

**8. left distributivity of * over +: For every q, s, and t in R, q*(s+t)=q*s+q*t**

**right distributivity of * over +: For every q, s, and t in R, (s+t)*q=s*q+t*q**

The set of integers mod *m* is denoted *Z*_{m}. The elements are written [*x*]_{m} and are equivalence classes of integers that have the same integer remainder as *x* when divided by *m*. For example, the elements of [–5]_{m} are of the form –5 plus integer multiples of 7, which would be the infinite set of integers {. . . –19, –12, –5, 2, 9, 16, . . .} or, more formally, {*y*: *y* = -5 + 7*q* for some integer *q*}.

Modular addition, [+], is defined in terms of integer addition by the rule

[*a*]_{m} [+] [*b*]_{m} = [*a + b*]_{m}

Modular multiplication, [*], is defined in terms of integer multiplication by the rule

[*a*]_{m} [*] [*b*]_{m} = [*a * b*]_{m}

*Note: For ease of writing notation, follow the convention of using just plain + to represent both [+] and +. Be aware that one symbol can be used to represent two different operations (modular addition versus integer addition). The same principle applies for using * for both multiplications.*

**A. Prove that the set Z_{31}(integers mod 31) under the operations [+] and [*] is a ring by using the definitions given above to prove the following are true:**

**1. closure property of [+]**

**2. closure property of [*]**

**3. additive identity property**

**4. additive inverse property**

**5. associative property of [+]**

**6. associative property of [*]**

**7. commutative property of [+]**

**8. left and right distributive property of [*] over [+]**