Weakly factorial domains and groups of divisibility

Abstract:

An integral domain $R$ is said to be weakly factorial if every nonunit of $R$ is a product of primary elements. We give several conditions equivalent to $R$ being weakly factorial. For example, we show that the following conditions are equivalent: (1) $R$ is weakly factorial; (2) every convex directed subgroup of the group of divisibility of $R$ is a cardinal summand; (3) if $P$ is a prime ideal of $R$ minimal over a proper principal ideal ($\left ( x \right )$), then $P$ has height one and ${\left ( x \right )_P} \cap R$ is principal; (4) $R = \cap {R_P}$, where the intersection runs over the height-one primes of $R$, is locally finite, and the $t$-class group of $R$ is trivial.