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Let $\ast$ be a star-operation on $R$ and $\ast {}_s$ the finite character star-operation induced by $\ast$. The purpose of this paper is to study when $\ast =v$ or $\ast {}_s=t$. In particular, we prove that if every prime ideal of $R$ is $\ast$-invertible, then $\ast =v$, and that if $R$ is a unique $\ast$-factorable domain, then $R$ is a Krull domain.

Let $D$ be a principal ideal domain and be its self-idealization. It is known that $R(D)$ is a commutative noetherian ring with identity, and hence $R(D)$ is atomic (i.e., every nonzero nonunit can be written as a finite product of irreducible elements). In this paper, we completely characterize the irreducible elements of $R(D)$. We then use this result to show how to factorize each nonzero nonunit of $R(D)$ into irreducible elements. We show that every irreducible element of $R(D)$ is a primary element, and we determine...