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On the Wikibooks page Intermediate Algebra/Algebraic Axioms, the author included $0 \neq 1$ as the axiom that relates addition, and multiplication.

But, what I see the most is just the distributive law. What does $0 \neq 1$ mean? And how it is related to addition and multiplication? Why is it included in this list of axioms?

JOHN
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    https://en.wikipedia.org/wiki/Field_with_one_element may be worth a look. – Barry Cipra Aug 18 '19 at 23:49
  • It is a convention employed mainly for convenience to exclude the trivial one element ring from bein a field (which would be a degenerate counterexample to many theorems). This is discussed in many prior questions, e.g. see here and its linked questions. – Bill Dubuque Aug 18 '19 at 23:58
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    @BarryCipra That link will only serve to confuse beginners. $\Bbb F_1$ - the imagined "field" with one element - is not the same as the trivial ring with one element - which is what is being excluded in the definition of a field. – Bill Dubuque Aug 19 '19 at 00:02
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    "What does $0≠1$ means?" - I don't think I understand what you're asking. Are you just asking for the literal meaning of that statement (which is "zero is not the same thing as one")? Or is there something in particular about that statement that you're confused about? Are you asking about why that statement was included in the list of axioms? – Tanner Swett Aug 19 '19 at 02:38
  • I answered the question that was asked and my answer received several downvotes. Another user posted an answer that does not answer the question that was asked, and that answer received several upvotes, and you accepted it. So, I'm taking the liberty of editing this question to match the answer, since apparently you're satisfied with that answer. – Tanner Swett Sep 10 '19 at 16:06
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    @BarryCipra Did you miss my earlier comment? Do you really wish to confuse readers by linking to unrelated topics (one often wrongly confused as being related by beginners due to similar naming) – Bill Dubuque Nov 26 '19 at 23:38
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    @BillDubuque, I did see your comment. I agreed the link I recommended looking at could be confusing, but disagreed it could only be confusing. I thought your comment served to clear up the potential confusion, so I decided to leave the link for anyone who might find it of interest. – Barry Cipra Nov 27 '19 at 13:00
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    @Barry Surely it can only be confusing and misleading for someone who posed a question at this level, since one needs a deep understanding of abstract algebra to appreciate the matters discussed in your link. Those without such are often misled into believing (due only to its name) that it is somehow related to matters like this, when that is not true. This has happened frequently in the past in general level math forums, so it is sad to see someone insist on further propagating such confusion. Alas, comments cannot be downvoted and deleted, – Bill Dubuque Nov 27 '19 at 13:52
  • @Bar from Lorscheid's $\Bbb F_1$ for everyone "The first thought that crosses one’s mind in this context is probably the question: What is the “field with one element”? Obviously, this oxymoron cannot be taken literally as it would imply a mathematical contradiction [...] However, many approaches contain an explicit definition of $\Bbb F_1$, and in most cases, the field with one element is not a field and has two elements. Namely, the common answer of many theories is that $\Bbb F_1$ is the multiplicative monoid {0,1}, lacking any additive structure" – Bill Dubuque Nov 28 '19 at 04:20

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A ring $R$ where the additive identity and the multiplicative identity are the same is boring. $$\forall x\in R,x\cdot0=0$$ $$\forall x\in R,x\cdot1=x$$ So if $1=0$, then that unity is the only member of the ring.

So, should $\{0$} be a ring? Some authors say yes, and call it the zero ring. Others note that this non-interesting structure makes a lot of things harder to talk about down the road, and so they avoid having to say "Let a non-zero ring $R$" be given every time the zero ring would be a counterexample to a proposition. Those authors get around it by just introducing an extra axiom to ban the zero ring.

(In some way, it's similar to how middle school students will ask why the prime numbers are given a more complicated definition just to make it specific that 1 isn't prime. It's a bit hard to explain to them that that is a lot prettier than the way the Unique Factorization Theorem would have to get patched if 1 were prime, but that's the long and the short of it.)

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    Do you have any examples of theorems that do not apply to the zero ring? I remember taking an algebra course and marveling at how the professor introduced $0\neq 1$ as an axiom for rings, but then never used it even once. – Milo Brandt Aug 19 '19 at 00:22
  • The example that popped into my head off the cuff was that you wouldn't want to think of the zero ring as a subring of the integers. It might have an impact on the definition of a field as well, because {0} is definitely not a field. –  Aug 19 '19 at 00:25
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    The zero ring isn't a subring of the integers if you define "subring" so that a subring has to have the same additive and multiplicative identities as the original ring. – Tanner Swett Aug 19 '19 at 02:23
  • Agreed. My point is that that's the kind of disclaimer you don't have to worry about if you add a $0\neq1$ axiom to your definition. It's a stylistic choice, but one that I would be very sympathetic to if I were ever to write a book on abstract algebra. –  Aug 19 '19 at 02:33
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    Disallowing the zero ring is rarely adopted and a terrible idea; it is deeply misleading to lend credence to the idea that it "makes a lot of things harder to talk about down the road". This author is axiomatizing fields, not rings, where the requirement that $0\neq 1$ actually is natural and is universally accepted. – Eric Wofsey Nov 27 '19 at 16:44
  • @MatthewDaly No "disclaimer" is needed since it follows from the general definition of substructure. You have yet to give even a single example of "lots of things that are harder to talk about down the road". Do you know any authors whose definition of ring excludes the one element ring? – Bill Dubuque Nov 28 '19 at 02:49
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    Working with trivial objects can be tricky, even if in some sense unimportant, because they're often unintuitive, and it can take a while for the best definition to be clear in any case. Put this all together, and you can have a centuries-long debate about whether to include or omit a single example from a definition. (Well, as far as I know, only one such debate lasted longer than a century, which is the one over whether 1 is a prime number. But one example is enough to verify that it can happen.) So while Tanner, Eric, and Bill are right, let's not be too hard on Matthew about it. – Toby Bartels Dec 02 '19 at 18:51