## Description

This info is just for reference. I need A and B answered.

An integral domain *Z* is a ring for the operations + and * with three additional properties:

1. The commutative property of *: For any elements *x* and *y* in *Z*, *x***y*=*y***x*.

2. The unity property: There is an element 1 in *Z* that is the identity for *, meaning for any *z* in *Z*, *z**1=*z*. Also, 1 has to be shown to be different from the identity of +.

3. The no zero divisors property: For any two elements *a* and *b* in *Z* both different from the identity of +, *a***b*≠0.

A field *F* is an integral domain with the additional property that for every element *x* in *F* that is not the identity under +, there is an element *y* in *F* so that *x***y*=1 (1 is notation for the unity of an integral domain). The element *y* is called the multiplicative inverse of *x*. Another way to explain this property is that multiplicative inverses exist for every nonzero element.

Modular multiplication, [*], is defined in terms of integer multiplication by this 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 multiplication versus integer multiplication).*

*A. Prove that the ring Z_{31} (integers mod 31) is an integral domain by using the definitions given above to prove the following are true:*

*1. The commutative property of [*]*

*2. The unity property*

*3. The no zero divisors property*

*B. Prove that the integral domain Z_{31} (integers mod 31) is a field by using the definition given above to prove the existence of a multiplicative inverse for every nonzero element.*

## Explanation & Answer

all done! let me know if you need anything else :)

A. Prove that the ring Z31 (integers mod 31) is an integral domain by using the definitions

given above to prove the following are true:

1. The commutative property of [*]

The commutative property of *: For any...