Weakly factorial domains and groups of divisibility

Abstract

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