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Find elements $a,b$, and $c$ in the ring $\mathbb Z \oplus\mathbb Z \oplus\mathbb Z $ such that $ab$, $ac$, and $bc$ are zero divisors but $abc$ is not a zero divisor.

I am not sure how to approach this problem guidance is appreciated.

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Unless you are using some strange ring structure on the sum, this shouldn't be possible. If $abd=0$ for some nonzero $d$, don't you agree that $(abc)d=(abd)c=0$ as well? – rschwieb Dec 10 '13 at 21:00
Well I was just thinking that you could have a=(1,1,0); b=(1,0,1); c=(0,1,1) or -1 for all the ones. – MathematicalAnomaly Dec 10 '13 at 21:08
You can if you exclude $0$ as a zero divisor. – rschwieb Dec 10 '13 at 21:11
zero is a never a zero divider by the definition of a zero divider. – MathematicalAnomaly Dec 10 '13 at 21:22
If you define it that way, yes. Some people don't. It's fine if you do, but just be aware that definitions are not always universal (despite the hopes of some.) So it's always best to mention which way you want to go, if there is any doubt. – rschwieb Dec 10 '13 at 21:24
up vote 1 down vote accepted

The zero divisors of this ring are the elements that are zero in at least one place.

The product of two elements always has at least as many zeros as the factors. If the product of two elements has a zero then, a product of three elements certainly will have more.

Your example of $(1,1,0),(0,1,1),(1,0,1)$ will work provided you define away $0$ from being a zero divisor.

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