Originally Posted by

**Deveno**

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It is possible for elements of a ring $R \times S$ to be neither, although that does not happen if $R,S$ are finite cyclic rings. An example is given by $R = \Bbb Z, S = \Bbb Z_2$, where the element $(2,1)$ is neither:

(2,1)*(a,b) = (0,0) implies (2a,b) = (0,0), which implies b = 0, and 2a = 0. The only integer a for which 2a = 0, is 0, so (2,1) is not a zero-divisor.

(2,1)*(a,b) = (1,1) implies (2a,b) = (1,1), which implies b = 1, and 2a = 1. There is NO integer a for which 2a = 1 (1/2 is NOT an integer), so (2,1) is not a unit.