If I divide zero by any number, I get zero. Would that not make it composite, perhaps even infinitely composite?
Edit: I AM NOT ASKING IF IT IS PRIME. I AM ASKING IF IT IS INFINITELY COMPOSITE
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3What is composite? – zoli Feb 25 '17 at 22:02
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Opposite of prime. – Xetrov Feb 25 '17 at 22:08
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1Actually $0$ is prime: If $0\mid ab$ then $0\mid a$ or $0\mid b$. - But $0$ is not irreducible: It can be written as product of two non-units, e.g., $0=0\cdot 2$. – Hagen von Eitzen Feb 25 '17 at 22:12
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1@HagenvonEitzen I don't think it's at all standard to consider $0$ a prime. Otherwise, we'd have books full of statements like, "let $p$ be a non-zero prime". Maybe you are joking? – Théophile Feb 25 '17 at 22:17
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1@Théophile See Why doesn't $0$ being a prime ideal in $\Bbb Z$ imply that $0$ is a prime number? – Bill Dubuque Feb 25 '17 at 22:19
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It appears that this question is about convention and terminology, not about mathematical facts/truth/reality. To say it is about "what's the definition?" makes it even less interesting, since definitions change with time (as do conventions), and are usually somewhat-stilted incarnations of convention.
So, sure, every integer divides $0$, and this is true in any ring. The real point is that this makes the divisibility properties of $0$ both very special and very uninteresting. There are far more interesting and worthwhile examples of "infinite divisibility", for example in the ring of polynomials $\mathbb C[x,x^{1/2},x^{1/4},x^{1/8},x^{1/16},\ldots]$.
paul garrett
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