# 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 *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 *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 [+]