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Proving subring S=mZ

Let S be a subring of the integers Z. Prove that S=mZ={mx|x is an integer} for some m€lNu{0}.

I have had a go. Can someone tell me if I am lacking justification or if I have done a mistake please.
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Original post by cooldudeman
Let S be a subring of the integers Z. Prove that S=mZ={mx|x is an integer} for some m€lNu{0}.

I have had a go. Can someone tell me if I am lacking justification or if I have done a mistake please.
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Looks fine to me. (The question is pretty much "prove that Z is a principal ideal domain", by the way, if you recognise those terms. Don't worry if not.)
Original post by Smaug123
Looks fine to me. (The question is pretty much "prove that Z is a principal ideal domain", by the way, if you recognise those terms. Don't worry if not.)


Can I just ask why is the penultimate line true? I was just told to include that in the answer but I didnt understand.
Original post by cooldudeman
Can I just ask why is the penultimate line true? I was just told to include that in the answer but I didnt understand.

It's entirely equivalent to the following, which I would consider more natural:

"WLOG a > 0. Indeed, if it were not, continue with -a instead, since -a is in the subring by additive closure."
Original post by Smaug123
It's entirely equivalent to the following, which I would consider more natural:

"WLOG a > 0. Indeed, if it were not, continue with -a instead, since -a is in the subring by additive closure."


I think I got it. But what does WLOG mean?

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Original post by cooldudeman
I think I got it. But what does WLOG mean?

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Without loss of generality.

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