Is the Direct Sum of Two Nonzero Rings Ever an Integral Domain?

[kathrynag]

Show that the direct sum of 2 nonzero rings is never an integral domain


I started by thinking about what a direct sum is
(a,b)(c,d)=(ac,bd)
(a,b)+(c,d)=(a+c,b+d)
We have an integral domain if ab=0 implies a=0 or b=0

 

[snipez90]

Hint: The first property is pretty relevant. The second not so much.

 

[kathrynag]

So we look at (a,b)(c,d) with (a,b) not zero and (c,d) not zero. Then multiplying together will never result in 0

 

[Char. Limit]

What about a=d=1, and b=c=0? Then you get (1,0)*(0,1) = (0,0).