90 problems

Let $R$ be a ring and let $M$ be an $R$-module. A flat cover of $M$ is a flat $R$-module equipped with a cover morphism onto $M$. Enochs's flat cover conjecture. Every $R$-module h…

Let $A$ be a unital Banach algebra, and suppose that every maximal left ideal of $A$ is finitely-generated. Dales–Želazko conjecture. Then $A$ is finite-dimensional. This conjectur…

Throughout, let $R$ be a commutative ring with identity, let $S\subset R$ be a multiplicative subset, and let $R_S$ denote the localization of $R$ with respect to $S$. An $R$-modul…

Let $R$ be any ring. A module $F$ is flat when tensoring with $F$ preserves exact sequences, and a class of modules is covering when every module has a cover by an object of that c…

Let $M$ be a closed oriented $3$-manifold, and let $\mathcal S_{2,\infty}(M)$ denote its Kauffman bracket skein module over $\mathbb Z[A^{\pm 1}]$. Marché's decomposition conjectur…

Boyle's conjecture. Every left QI-ring is left hereditary.

Let $\bd_1,\bd_2,\ldots,\bd_d$ be the lattice diagram modules associated with the partitions obtained by removing the corner cells of a partition. Science Fiction conjecture. The m…

Let $R$ be a non-Gorenstein local ring, and suppose that there is a nonfree totally reflexive $R$-module. The Takahashi conjecture. There are infinitely many isomorphism classes of…

Direct-sum-of-uniforms conjecture. $E$ is a direct sum of uniform modules.

Throughout, let $p$ be a prime, let $C_p={\bold Z}/p{\bold Z}$ with generator $\alpha$, and let $R={\bold Z}C_p$. For an $R$-module $M$, let $\pi_M:{\bold Z}M\to M$ be the canonica…

Fix $m\geq 0$. Let $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $A_m^{(n)}$ be the set of $(n+1,m)$-excellent polynomials, meaning polynomials $f$ for which $\delta_{n+1}(f)$ is a polynom…

Ascending chain property. Every oligomorphic structure has the ascending chain property: all orbit-finitely spanned vector spaces over the structure are Noetherian.

Uniform generation conjecture. There is a $k\in\mathbb{N}$ such that $\mathcal O_{\mathbf{A},\mathbf{R}}$ is uniformly generated by $k$-ary $(\mathbf{A},\mathbf{R})$-minors.

Let $\Lambda$ be a finite-dimensional algebra. A module $M\in\operatorname{mod}\Lambda$ is a brick if $\operatorname{End}_{\Lambda}(M)$ is a division algebra. The algebra $\Lambda$…

Let $A_n$ be the $n$th Weyl algebra, and let a finitely generated $A_n$-module be torsion when it has rank zero. A module is cyclic if it is generated by one element. Stafford's co…

Let $\Bbbk\mathbf{gr}^{\operatorname{op}}$ be the opposite-category module category associated with the handlebody Jacobi-diagram category, and let $A^{L}_d Q$ denote the submodule…

Relative flaprojective periodicity conjecture. The $A$-modules of cocycles of $F^\bullet$ are $A/R$-flaprojective.

Let $\widehat{\mathfrak{b}}$ be the algebra whose modules are under consideration, and let $\widehat{V}$ and $\widehat{W}$ be the $\widehat{\mathfrak{b}}$-submodules appearing in t…

Let $P$ be a finite non-abelian $p$-group. Write $\Phi(P)$ for its Frattini subgroup, $Z(\Phi(P))$ for the center of that subgroup, and say that a module is cohomologically trivial…

Let $A\to B$ be the DG algebra homomorphism in the setting preceding the conjecture, and let $N$ be a DG $B$-module. Write $\operatorname{\texttt{(AR1)}}$ and…

All rings are commutative with identity, and all modules are unitary. A module $P$ over a ring $R$ is a Baer module if ${\rm Ext}_R^1(P,T)=0$ for every torsion $R$-module $T$. A ri…

Let $D$ be a commutative domain. A $D$-module $P$ is a Baer module if ${\rm Ext}_D^1(P,T)=0$ for every torsion $D$-module $T$. Kaplansky's conjecture. Every Baer module over a comm…

Proposed conjecture. The ring $R$ is $\tau_q$-von Neumann regular if and only if every semi-regular flat $R$-module is flat.

Torsion analogue conjecture. A tight submodule of a direct sum of cyclically presented torsion modules in ${\mathcal C}_V^*$ should itself be a direct sum of cyclically presented m…

Converse Wakamatsu tilting conjecture. If $|T|=|\Lambda|$, then $T$ is Wakamatsu tilting.