25 problems

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 $D$ be a Prüfer domain. A locally principal ideal of $D$ is an ideal that becomes principal upon localization at every maximal ideal, and $D$ is of finite character if every no…

Polini–Ulrich conjecture. One has

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 $(A,\mathfrak{m})$ be a two-dimensional regular local ring, let $I$ be an $\mathfrak{m}$-primary ideal, and let $\overline{v}_I$ denote the asymptotic Samuel function of $I$. W…

Core formula conjecture. The core of $I$ is

Let $R$ be a domain. It is pseudo-integrally closed when $R=\widetilde{R}=\bigcup(I_t:I_t)$, where $I$ ranges over the finitely generated ideals of $R$. A Krull-type domain is a Pr…

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 $D$ be a strongly discrete Prüfer domain, and let $\mathrm{Inv}(D)$ denote its group of invertible ideals. Freeness conjecture. If $D$ is a strongly discrete Prüfer domain, the…

Let $\mathcal{W}$ be the van der Waerden ideal and let $\mathcal{I}_{1/n}$ be the summable ideal associated with the sequence $(1/n)_{n\in\mathbb{N}}$. The notation…

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…

Corso–Polini–Ulrich's core formula conjecture.

Failure conjecture. The quotient

Let $D$ be an integral domain. The constructible topology on the space of ideals is denoted by $\mathcal{I}(D)^{\mathrm{cons}}$, and $\mathrm{Princ}(D)$ denotes the set of principa…

A Bézout monoid is an abstract divisibility monoid of the type arising from finitely generated ideals of a Bézout ring. For a commutative unital ring $R$, let…

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 $\mathscr{T}\colon \operatorname{KHaus}\to\operatorname{CMSLat}$ assign to a compact Hausdorff space its lattice of closed sets, ordered by reverse inclusion, and let…

Clifford t-class semigroup conjecture. If $R$ is a $v$-domain, then $R$ has Clifford $t$-class semigroup if and only if $R$ is a Krull-type domain.

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…

Let $R$ be a ring, and let $G(R)$ denote its intersection graph of ideals. A graph is weakly perfect when its clique number equals its chromatic number, that is,…

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

Let $D$ be an integral domain. An H-domain is a domain in which every $t$-maximal ideal is divisorial, and an ideal is faithfully flat when it is faithfully flat as a $D$-module. G…

Let $A$ be an integral domain. An H-domain is a domain in which $t$-maximal ideals are divisorial. Glaz–Vasconcelos conjecture. In an H-domain, faithfully flat ideals should be pro…

Let $R$ be a local Gorenstein ring with infinite residue field, let $I$ be an $R$-ideal with $g=\operatorname{ht} I>0$ and $\ell=\ell(I)$. Assume $I$ satisfies $G_{\ell}$ and … Let…