This question confuses me in many ways, by "non-zero" is it meant a ring with no divisors of zero?
As an ID is a commutative ring-with-a-1 no divisors of zero, I guess i'm going to have to show there does exist a divisor of zero in R1 X R2?
Not sure at all how to go about starting this one...
It depends on how you define rings. Some authors insist that a ring must contain a nonzero multiplicative identity. Others do not insist on this condition. If you use the second alternative, then it is possible for a ring to contain just one element, namely the zero element.
A nonzero ring would then be a ring that contains at least one nonzero element.
Here’s a hint to get you started. Let be a nonzero element in and be a nonzero element in Then and are nonzero elements in Now multiply them together. What do you get?
You would get (0,0) in R1 X R2 which would imply this ring has a element, say Q, such that Q multiplied by any element = Q, so it isn't an integral domain?
Obviously i'd write out the proof more concise than, but am I getting at the right idea?
What is the definition of an integral domain?
Originally Posted by pkr
Commutative ring with a 1, no divisors of zero.
So, because a1 and a2, are non-zero elements of R1 and R2, and R1 X R2 = (0, 0) implies R1 X R2 does posses a divisor of zero, and thus cannot be a integral domain.