- Let denote the ring of integers modulo . Then does not satisfy the zero product property: 2 and 3 are nonzero elements, yet .
- In general, if is a composite number, then does not satisfy the zero-product property. Namely, if where, then and are nonzero modulo, yet .
- The ring of 2 by 2 matrices with integer entries does not satisfy the zero-product property: if
- and ,
- yet neither nor is zero.
- The ring of all functions, from the unit interval to the real numbers, has zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions, none of which is identically zero, such that is identically zero whenever .
- The same is true even if we consider only continuous functions, or only even infinitely smooth functions.
Read more about this topic: Zero-product Property
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