9 problems

Samuel's conjecture. The ring $R[[t]]$ is a unique factorization domain.

Catenarity conjecture. Every Noetherian unique factorization domain, and more generally every integrally closed domain, should be catenary.

Wolffhardt's conjecture. If $\operatorname{Sym}_R(M)$ is a $\operatorname{UFD}$, then

A local unique factorization domain is a local domain in which every nonzero nonunit factors uniquely into irreducibles. A local ring is catenary when saturated chains of prime ide…

Compressed-graph invariance conjecture. Under these hypotheses,

Compressed-graph structure conjecture. An edge joins the vertices represented by $\"]\phi(\prod_i d_i)$ and $\"]\phi(\prod_i b_i)$ if and only if

GCD-representative conjecture. For every $a\in D$,

Extension conjecture. The results obtained for quotient rings of unique factorization domains should extend to any quotient ring of a unique factorization domain.

Unique factorization obstruction conjecture. For acyclic cluster algebras, the existence of a seed with respect to which some exchange polynomial is reducible or two exchange polyn…