Rings proving associativity

Trying to see if IN is a ring. Have I done everything correctly?

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The answers should not start "Assume N is a ring", where N is the naturals or the naturals with zero; what you're trying to do is to prove that N is a ring (or not), so assuming it tells you nothing. The rest of your reasoning is fine, although it's a bit unclearly phrased. I'd say something like "Do additive inverses exist? That is, is it true that for all x there is y such that x+y=0? No; there isn't even a zero in this set."

In order to show that something is not a ring, it is enough to show that there is no additive identity. That is, there is no element $e$ of the set such that for all x, $e+x = x$. This is clearly true for the naturals, since $e+x > x$ for all e and for all x in the naturals.

In order to show that something is not a ring, it is also enough to show that some element has no additive inverse. This is clearly true for the naturals-with-zero, since $1+x > 1 > 0$ for all $x \not = 0$, so 1 cannot have an inverse.

In general, associativity is the last property you ever want to show, because it's annoying and a bit fiddly - lots of unenlightening symbols to write down. It's usually easiest to demonstrate an element which has no (additive) inverse.

ok I understand but how would I prove associativity because im guessing I will have to at some point, like in part 1c.

correct me if im wrong:
in 1a, its not a ring since there is no special element 0 in natural numbers, as you mentioned already.
in 1b, its not a ring since there is not an additive inverse for all elements, giving a counter example, 4 is an element in N_0 but there does not exist an element y in N_0 such that 4+y=0=y+4.

1c seems like a ring to me but I don't know how to prove it.

To my mind, nearly every thing you "prove" is basically you just restating what it is you need to prove and writing a tick beside it. It's not terribly convincing.

To be fair, actually proving these results for something like $\mathbb{N}$ is quite tricky, and probably requires material you haven't covered yet, But you should at least recognize writing "$\forall x,y \in \mathbb{N}, \, x+y \in \mathbb{N}$ (tick)" doesn't really prove anything.

Those are correct, if you prove that no such y exists. (It's easy: 4+y >= 4 > 0, and in particular is not 0.)

You haven't defined the example of 1c.

Sorry here it is.
For 1c can I just say we know that Z is a ring so 3Z is a ring.

What reasoning have you performed there? (It's false: 3Z doesn't have a multiplicative identity. 3Z is an ideal of Z, though.)

But is that axiom required for a ring? This is the definition of rings we have and it doesn't include the multiplicative identity.

Because the question says which are 'rings' and not which are 'rings with one'.

Ah, sorry - my course defined "ring" as "commutative ring with one".

In that case, yep, 3Z is a ring. What reasoning did you use?

I don't know the reasoning. My guess is that we use the fact that Z is a ring. But I really don't know.

In that case, you might like to prove that "an ideal of a ring is a ring". That would be reasoning enough.

Is that based on a theorem? What is an ideal of a ring? Like a function of a ring?

Oh, I see. An ideal is a bit like a sub-ring, but more useful - you'll meet it soon, if you haven't already.

OK, in that case, you're going to have to prove directly that 3Z is a ring, by showing that it is a group, and showing that the multiplicative properties hold. (Your job is made easier by the fact that most of the multiplicative properties of 3Z are equivalent to the properties in Z.)

Ok here is what I have for 1c. Have I missed anything out?

Also for 1d, id say that it is a ring but do I have to prove it?

And for 1e, it is not a ring because it is not always closed under multiplication since for example, (X^5)*(X^5) is not an element of 1e?

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Yeah, that's probably fine.

Yes, you must prove 1d. (It's weird for me that "ring" for you is "commutative ring not necessarily with one"…)

Yes, 1e is correct.

With 1c, how do I know which axioms that don't need proving?

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Well, technically you need to prove them all. It's just that there are some which are really horrible to prove (things like (a+b)+c = a+(b+c) in the reals). You're usually free to assume properties of things like Z as long as you state them.