15 problems

Let $R$ be a commutative unitary ring. Write $J(R)$ for its Jacobson radical, and let $\textsc{I}(\mathcal{S}_R)$ denote the Erdős–Burgess constant of the multiplicative semigroup…

Let $R$ be a finite commutative local ring with identity, of cardinality $32$, and suppose that $R$ is not a field. Let $\Gamma(R)$ be the simple graph whose vertices are the nonze…

Let $R$ be a commutative ring with non-zero unity, and let $\Gamma(R)$ be the graph whose vertices are the non-zero zero-divisors of $R$, with distinct vertices adjacent exactly wh…

Let $R$ be a commutative ring and let $P$ be a Col-divisible polytope of arbitrary dimension. Write $K_i(R,P)$ for its associated $K$-groups, and let ${\mathfrak c}(P)\leq\dim P$ b…

Let $R$ be a commutative ring with identity. For a positive integer $s$, let $p_{2s}(R)$ denote the least number of $2s$-th powers needed in the relevant representation defining th…

Let $\mathcal{R}$ be a commutative ring with unity, and let $n\in\mathcal{R}$. A Diophantine quadruple with property $D(n)$ is a set…

Weighted Erdős–Burgess lower-bound conjecture.

Let $A$ be a commutative ring with unit in which $2\neq0$, let $(N,Q)$ be a quadratic module over $A$, and set $\mathcal{C}:=\mathcal{C}(N,Q)$. Let $e$ be the idempotent used to fo…

Let $R$ be a commutative ring and let $(M,q)$ be a quadratic module over $R$. Write $M^{\perp}$ for the radical of the polar form of $q$, and let an orthogonal map mean an $R$-line…

Variable-exponent central power-difference conjecture. Under these conditions, $R$ is commutative.

Non-isomorphism conjecture. There exists a commutative ring $\Bbbk$ such that the functors $H_*^{\sf RT}$ and $H_*^{\sf RS}$ are non-isomorphic.

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…

Let $\mathbf U$ be the quantum group and let $\mathbb P=P\otimes_R\mathcal F$ be as above, where $\mathcal F$ is a finite-dimensional $\mathbf U$-module. Let…

Nilpotence conjecture. One has

Total-graph coloring conjecture.