Two counterexamples in abstract factorization

Abstract

We give examples that provide negative answers to three questions about abstract factorization posed by Anderson and Frazier. We show that (1)~an atomic domain need not be $\tau$-atomic for $\tau$ divisive, (2)~an atomic domain need not be a comaximal factorization domain (CFD) and (3)~for $\tau$ divisive, a nonzero nonunit of a $\tau$-UFD need not be a $\tau$-product of $\tau$-primes. Along the way, we generalize the theorem of Anderson and Frazier that a UFD is a $\tau$-UFD for $\tau$ divisive (with a simplified proof), and we demonstrate a method for constructing domains with no pseudo-irreducible elements.