abstract algebra
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
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 Zm. 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 + 7q 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 Z31(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 [+]

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