Proving that the ring Z_31 (integers mod 31) is an integral domain and a field

[WVteacher]

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
[*]
2. The unity property
3. The no zero divisors property

B. Prove that the integral domain Z₃₁ (integers mod 31) is a field by using the definition given above to prove the existence of a multiplicative inverse for every nonzero element

I have tried to prove these but I cannot seem to get what the graders are looking for.

 

[HallsofIvy]

So what answers did you give and what did the grader say about them?

Since you give the requirements for an integral domain as
"1. The commutative property of [*]
2. The unity property
3. The no zero divisors property"

How did you try to prove those are true for \(\displaystyle Z_{31}\)?
How did you try to prove that ab (mod 31)= ba (mod 31)?
Is there a member of \(\displaystyle Z_{31}\), e, such that ex= xe= x for any x in \(\displaystyle Z_{31}\).
How did you try to prove that xy is not 0 (mod 31) for any x and y in \(\displaystyle Z_{31}\).

The second part asks you to show that this same system is a field. Do you know the difference between an "integral domain" and a "field"?
The fact that 31 is a prime number is important here.

(Strictly speaking, in order to prove that a system is an integral domain or field you also need to prove that it forms a group under addition. Have you already done that or are you not required to do that?)

 

[WVteacher]

For the commutative I proved that [a]31*31=31*[a]31 with the correct steps in between.
the unity property is difficult for me to prove because I showed that a=a*b=b. Then I finished by stating that the multiplicative identity is 1 so Z31*1=Z31. I also stated that a and b are units of Z. They told me that this was not a proof. So I am not sure how to approach this.
The non zero property was proven by stating that 31 is prime so that a*b will not divide 31 so that a=0 or b=0. This means that Z31 has no zeros. they told me that I needed more justification.

Proving the integral domain is a field I stated the multiplicative inverse but was told that this was not the correct approach.

 

[HallsofIvy]

For the commutative I proved that [a]31*31=31*[a]31 with the correct steps in between.

So you got this right? Good!

the unity property is difficult for me to prove because I showed that a=a*b=b.

What? You don't say what "a" and "b" are. If you mean that "a" and "b" are any members of Z31 it simply is not true. The only "a" and "b" such that "a= a*b= b" are a= b= 1. The "unity property" is "there exist a in Z31 such that, for any b in Z31, ab= b.

Then I finished by stating that the multiplicative identity is 1 so Z31*1=Z31.

Yes, you prove that Z31 has an identity by showing that 1*a= a*1= a for all a in Z31.

I also stated that a and b are units of Z. They told me that this was not a proof. So I am not sure how to approach this.

Again, what are "a" and "b"? If they are to be "any members of Z31" then, again, that is not true. "0" does not have an inverse. What you want to prove is that "if a is not 0, there exist b, not 0, such that ab= 1".

The non zero property was proven by stating that 31 is prime so that a*b will not divide 31 so that a=0 or b=0. This means that Z31 has no zeros. they told me that I needed more justification.

Yes, what does "a or b do not divide 31" (NOT "a*b") have to do with a*b not being 0 (mod 31)? Where did you use the definition of Z31?

Proving the integral domain is a field I stated the multiplicative inverse but was told that this was not the correct approach.
They told me that this was not a proof. So I am not sure how to approach this.

You keep saying you stated things. You need to prove that those statements are true!