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is there any group or some numbers of math object, number or matrix that satisfies the following:

whenever $AB=0$ where A and B are math objects, some product of the "group" elements, out of them containing A and B, become zero.

edit: so, $ACDBE$ becomes zero.

nonzero matrices assumed.

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    Yes. $0 \times 0 = 0$, and $0 \times 0 \times 0 = 0$.2012-10-26

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Let $n$ be a composite number, say, $n=ab$ with $a\gt1,b\gt1$; then in arithmetic modulo $n$, you have $ab=0$, and any product of elements that include an $a$ and a $b$ will be zero.