A non-zero element x in a ring such that there exist non-zero y, z with yx = 0 = xz. There are no zero-divisors within ℤ, ℝ, or ℂ; generally, a commutative ring with an identity which has no zero-divisors is called an integral domain. For example, if A $=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$ and B $=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$ then AB = BA = 0, and so A and B are zero-divisors within the ring of 2 × 2 matrices. Factorization cannot be applied when zero-divisors exist; for example, the equation (x−1)(x + 1) = 0 over ℤ₈ does not imply that x = 1 or x = –1, as x = 3 and x = 5 are also solutions because 2 × 4 = 4 × 6 = 0 in ℤ₈.