Show that the direct sum of two nonzero rings is never an in

Show that the direct sum of two nonzero rings is never an integral domain.

Solution

solution-

Let A and B be non zero rings.

Then

there exist non zero elements a A and b B.

Then

(a, 0) and (0, b) are non zero elements of A B,

but

(a, 0)(0, b) = (0, 0),

thus

A B is not an integral domain.