Some examples of weakly factorial rings

Abstract

Let $D$ be a principal ideal domain, $X$ be an indeterminate over $D$, $D[X]$ be the polynomial ring over $D$, and $R_n = D[X]/(X^n)$ for an integer $n \geq 1$. Clearly, $R_n$ is a commutative Noetherian ring with identity, and hence each nonzero nonunit of $R_n$ can be written as a finite product of irreducible
elements. In this paper, we show that every irreducible element of $R_n$ is a primary element, and thus every nonunit element of $R_n$ can be written as a finite product of primary elements.