13 problems

Core formula conjecture. One has

Let $L_k(n_1,n_2,\ldots,n_k)$ be the lattice considered in the paper, and let $I_{L_k(n_1,n_2,\ldots,n_k)}$ denote its join-meet ideal in the associated polynomial ring. Crystal co…

Let $k$ be a $\sigma$-field and $R$ a finitely $\sigma$-generated $k$-$\sigma$-algebra. A $\sigma$-ideal is radical if it is radical as an ordinary ideal, and mixed if, whenever…

Core formula conjecture. The core of $I$ is

The paper considers the ideal-theoretic identities established above for a commutative ring and natural number $n$. General ideal-lattice formula conjecture. The theorem above hold…

Let $\mathbb{k}$ be a field, let $R=\mathbb{k}[x_1,\ldots,x_d]$, and let $I\subseteq R$ be a radical homogeneous ideal of bigheight $h$. Harbourne's conjecture. One has … for every…

Let $R$ be a commutative ring. Say that $R$ has strong avoidance if every pair of ideals of $R$ whose union is $R$ has one ideal equal to $R$. For each prime ideal $\mathfrak{p}$ o…

The invariant partial ideal conjecture. A partial ideal of $\mathcal{A}$ arises from a total ideal if and only if it is an invariant partial ideal. Consequently, the map…

Let $W_+$ be the positive Witt algebra over a field of characteristic zero, and let ${\rm U}(W_+)$ be its universal enveloping algebra. The two-sided ideals of ${\rm U}(W_+)$ satis…

Let $k$ be a $\sigma$-field and $R$ a finitely $\sigma$-generated $k$-$\sigma$-algebra. A prime … of $R$ is minimal if every prime … with $\subset$ satisfies $=$. Minimal-prime fin…

Primary decomposition conjecture. For all $\mathbf{a}\in\mathbb{N}^n$,

The cyclotomic ideal-power conjecture. This equation has no solution except in the trivial cases

Polini–Ulrich conjecture. One has