Algebra: ring theory. Lecture 13, math 100 b, UCSD

For an integral domain D where every non-zero non-unit can be written as a product of irreducible elements, is a UFD if and only if every irreducible is prime. In every integral domain, primes are irreducible. PID implies UFD, and so Z, F[x] where X is a field, Z[i] , and Z[w] where w is a third root of unity, are UFDs. An ideal I of a unital commutative ring A is prime if and only if A/I is an integral domain.

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