A Closer Look at Non-unique Factorization via Atomic Decay and Strong Atoms

An elementary argument shows that a prime element must be irreducible, but the converse fails (see equation (∗) and its accompanying description in Section 4). Moreover, it is easy to verify that a prime element p has the nice property that any atom which divides a power pk of p must be an associate of p. We call an atom which has this property a strong atom (in the literature, these atoms have also been called absolutely irreducible [1] or [6, Definition 7.1.3], or completely irreducible [12]). An atom which is not strong must have a power which is divisible by a nonassociated atom. That is, powers of atoms can decay into other atoms, a phenomenon we refer to as atomic decay. More precisely, we have the following definition.