Suppose p is a prime. Find all the zero divisors in ×
Any member of is of the form (a, b) where a is in , b is in and multiplication is coordinate wise. That is, to be a zero divisor we must have for n and m integers. Since p is prime, at least one of a or c is a multiple of p and so is congruent to 0 modulo p. Either one of b or d is a multiple of , and so congruent to 0 modulo or both b and d are multiples of p.
I don't know what you mean by "the first set" and "the second set". There is only one set to be found here: the zero divisors of . Yes, since p is a prime, has no zero-divisors. So any zero-divisor of must be of the form (0, a) where a is a zero-divisor of . What are the zero-divisors of that?