Zero Divisors in Direct Product of Rings [duplicate]

abstract-algebra

For any $a\ne 0$, $(a,0)\cdot (0,a)=(0,0)$.


Hmmm...what about the elements $\,(1,0)\,\,,\,\,(0,1)\,$?

Of course if one, or both, of the rings have no unit you can choose any non-zero elements instead of $\,1\,$

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