10 problems

Let $D$ be a Dedekind domain of characteristic zero with quotient field $K$. The Dedekind-domain containment conjecture. Every idempotent $D$-plethory is contained in $K[e]$, or, e…

Half-factorial generating set conjecture. Every abelian group has a half-factorial generating set.

Half-factorial class-group conjecture. For every abelian group $G^*$, there exists a half-factorial Krull monoid, equivalently a half-factorial Dedekind domain, whose class group i…

Let $G$ be an abelian group. A subset $G_0\subseteq G$ is half-factorial if the monoid of zero-sum sequences over $G_0$ is half-factorial; it is a generating set if it generates…

Let $G$ be an abelian group. A Krull monoid is half-factorial if every nonunit has all its factorizations of the same length; a Dedekind domain is half-factorial when its multiplic…

Let $R$ be a Dedekind domain, let $G$ be a finite group, and let $G\to \operatorname{GL}_n(R)$ be an $R$-representation. Write $R[x_1,\ldots,x_n]_d^G$ for the degree-$d$ component…

Let $G$ be an abelian group. A commutative Dedekind domain is half-factorial if every nonzero nonunit can be written as a product of irreducibles and the number of irreducibles in…

Semistar-counting conjecture. The domain $D$ is a principal ideal domain (PID) if and only if

Let $(R,\mathfrak{m})$ be a local Dedekind domain with finite residue field, and let $n_1,\ldots,n_l$ be positive integers. For the finite $R$-module … write $\phi(M)$ for its cose…

Let $R$ be a Dedekind domain satisfying the assumptions above, let $\mathfrak{m}_i$ be maximal ideals of $R$, and let $n_i$ be positive integers. Let $M$ be the finite direct sum ……