sentences:  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