Prove a ring: showing an element has an additive inverse

[Bruce Wayne1]

I'm working on showing that Z₃ is a ring. The one portion I'd like to confirm is the additive inverse part. So here's what I'm thinking as my proof:

Given [x]₃ , suppose [3-x]₃ is the additive inverse in the set Z₃ . Thus:

[3-x]₃ =

[3]₃ + [-x]₃ =

[0]₃ + [-x]₃ =

[0-x]₃ =

[-x]₃
​

Then, it can be shown that

[x]₃ + [-x]₃ = 0

[x+ -x]₃ = 0​

Therefore, the additive inverse condition of a ring is met for Z₃.

 

[caffeinemachine]

I'm working on showing that Z₃ is a ring. The one portion I'd like to confirm is the additive inverse part. So here's what I'm thinking as my proof:

Given [x]₃ , suppose [3-x]₃ is the additive inverse in the set Z₃ . Thus:

[3-x]₃ =

[3]₃ + [-x]₃ =

[0]₃ + [-x]₃ =

[0-x]₃ =

[-x]₃
​

Then, it can be shown that

[x]₃ + [-x]₃ = 0

[x+ -x]₃ = 0​

Therefore, the additive inverse condition of a ring is met for Z₃.

Hello Batm.. oh sorry, I mean Mr. Wayne.

I am not sure what you want to do here. Do you want to show that in $\mathbb Z_3$ additive inverses of each element exist or do you want to show that $[x]_3$ has $[-x]_3$ as its additive inverse in $\mathbb Z_3$.

 

[Bruce Wayne1]

Hi!

I'm trying to show that for each element x ∈ R, there is a unique element y ∈ R such that x + y = y + x = 0. (denote y by −x.)

 

[caffeinemachine]

Hi!

I'm trying to show that for each element x ∈ R, there is a unique element y ∈ R such that x + y = y + x = 0. (denote y by −x.)

Okay.
We write $R=\mathbb Z_3$.
For $x\in R$, we want to show that:

1) Existence: $\exists y\in R$ such that $x+y=0$.
I think you did good. Only problem is that you began with "let $[3-x]_3$ be the additive inverse". I suggest you should have rather started with "we claim that $[3-x]_3$ is a candidate for $y$". Do you see why I say this?

2) Uniqueness: $x+y_1=0, x+y_2=0\Rightarrow y_1=y_2$.
You have not attempted to show the uniqueness of $y$. Try it out. Its not difficult. Post your attempt at this here and I'd be happy to comment/help.

Give my regards to the Bat if you ever meet him. BIG FAN!

 

[Bruce Wayne1]

Okay.
We write $R=\mathbb Z_3$.
For $x\in R$, we want to show that:

1) Existence: $\exists y\in R$ such that $x+y=0$.
I think you did good. Only problem is that you began with "let $[3-x]_3$ be the additive inverse". I suggest you should have rather started with "we claim that $[3-x]_3$ is a candidate for $y$". Do you see why I say this?

2) Uniqueness: $x+y_1=0, x+y_2=0\Rightarrow y_1=y_2$.
You have not attempted to show the uniqueness of $y$. Try it out. Its not difficult. Post your attempt at this here and I'd be happy to comment/help.

Give my regards to the Bat if you ever meet him. BIG FAN!

Thanks!

I have seen different proofs use "claim", and I do see that it makes a difference, though I didn't think of that here. Could you explain to me a bit better why claim is different than suppose in math proofs?

Truth is, I did terrible in high school geometry, and the math courses I took in college never gave me any real direction in proof writing. It was only in talking with a professor friend of mine that he mentioned proofs are the way to really learning math. And I've been watching lectures online and proving things out of textbooks, and reading forums.

Also, I didn't realize I needed to show uniqueness. Let me give it a try, and I'll post it. I'm working also on some other abstract algebra proofs. Maybe you could help me on those, too!

By the way, I haven't met the Batman yet, but if ever I do, I sure will mention you :)

 

[caffeinemachine]

Thanks!

I have seen different proofs use "claim", and I do see that it makes a difference, though I didn't think of that here. Could you explain to me a bit better why claim is different than suppose in math proofs?

Yeah sure.
When you say that "suppose $[3-x]_3$ is the additive inverse of [x]_3" then you are making two mistakes.
One, you are implicitly assuming that an additive inverse to $[x]_3$ exists. Further, even if existence is known, you cannot say that "suppose blah blah is an additive inverse of blum blum". You may say "we show that blah blah is an additive inverse to blum blum".
It makes no sense to first supposing that blah is additive inverse to blum and then then proving that blah is additive inverse to blum.
You may have more questions on this which I'd be happy to answer.

Truth is, I did terrible in high school geometry, and the math courses I took in college never gave me any real direction in proof writing. It was only in talking with a professor friend of mine that he mentioned proofs are the way to really learning math. And I've been watching lectures online and proving things out of textbooks, and reading forums.

You are doing a good job. It requires perseverance. Make sure that you are very clear about the definitions and notation. Intuition is very important to solve problems but the proofs should be rigorous and should not have things like "it is intuitively clear that" since intuition can be erroneous too.

Also, I didn't realize I needed to show uniqueness. Let me give it a try, and I'll post it. I'm working also on some other abstract algebra proofs. Maybe you could help me on those, too!

Sure! I am looking forward to solving your doubts.
A little tip from my side. If you are studying abstract algebra you might want to understand how the Principle of Mathematical Induction(PMI), especially the strong form, works. The Extremal Principle (which is PMI in disguise) is also very powerful. These two are basically incarnations of the Well Ordering principle. These principles are ubiquitous in Abstract Algebra and Combinatorial Mathematics.
Apart from this you might also occasionally encounter the surprising Pigeon Hole Principle. You may see some sample problem on this too.

By the way, I haven't met the Batman yet, but if ever I do, I sure will mention you :)

Thanks so much!.. (rubs eyes)