Atomic Domain - Special Cases

Special Cases

In an atomic domain, it is possible that different factorizations of the same element x have different lengths. It is even possible that among the factorizations of x there is no bound on the number of irreducible factors. If on the contrary the number of factors is bounded for every nonzero nonunit x, then R is a bounded factorization domain (BFD); formally this means that for each such x there exists an integer N such that x = x₁ x₂ ... x_n with none of the x_i invertible implies n < N.

If such a bound exists, no chain of proper divisors from x to 1 can exceed this bound in length (since the quotient at every step can be factored, producing a factorization of x with at least one irreducible factor for each step of the chain), so there cannot be any infinite strictly ascending chain of principal ideals of R. That condition, called ascending chain condition on principal ideals or ACCP, is strictly weaker than the BFD condition, and strictly stronger than the atomic condition (in other words, even if there exist infinite chains of proper divisors, it can still be that every x possesses a finite factorization).

Two independent conditions that are both strictly stronger than the BFD condition are the half factorial domain condition (HFD: any two factorizations of any given x have the same length), and the finite factorization domain condition (FFD: any x has but a finite number of non-associate divisors). Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization.