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 [+]