In this section, a ring can be viewed as merely an abstract set in which one can perform the operations of addition and multiplication; analogous to the integers.
Since a unique factorization domain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered.
Therefore, it is also natural to ask under what conditions particular elements of a ring can be "decomposed" without requiring uniqueness.
If every non-zero non-unit x of R can be written as a product of irreducible elements, R is referred to as an atomic domain.
If on the contrary the number of factors is bounded for every non-zero non-unit x, then R is a bounded factorization domain (BFD); formally this means that for each such x there exists an integer N such that if x = x1x2...xn with none of the xi invertible then 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 the 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[3]).