Zero-product Property - Non-examples

Non-examples

• 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 ,

then

,

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.