Exercise
When does the product R × R' contain zero divisors?
Never.
Only if R or R' has zero divisors.
Only if both R and R' have zero divisors.
Always.
How about (0,1') * (1,0') ?
No: (0,1') * (1,0') = (0,0').
No: (0,1') * (1,0') = (0,0').
Right!