On primary factorizations

AbstractWe relate ideals in commutative rings which are products of primary ideals to ideals with primary decompositions. Invertible primary ideals are shown to have special properties. Sufficient conditions are given for a primary product ideal to have a unique product representation. A domain is weakly factorial if every non-unit is a product of primary elements. If R is weakly factorial, Pic(R)=0. A Noetherian weakly factorial domain R is factorial precisely when R is integrally closed. R[X] is weakly factorial if and only if R is a weakly factorial GCD domain. Properties of weakly factorial GCD domains are discussed