A basic property of primes

Let R be a commutative ring and let p,q \in R be primes such that p does not divide q.. Show that if a \in R such that p|a and q|a, then pq|a.


Suppose p|a and q|a; then there exists t \in R such that qt = a. In particular, p|qt. Since p is prime in R and p does not divide q, p must divide t. Say ps = t. Then we have pqs = qt = a, and so pq|a.

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