By definition, a product of two primes has atwo factors (other than 1 and itself) and so it cannot be prime. So the set is not closed.
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No.
No.
No. The product of two primes cannot, by definition, be a prime.
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
Quite simply, they are closed under addition. No "when".