38 problems

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

Kaplansky's conjecture. $R$ is von Neumann regular if and only if $R$ is semiprime and each prime factor ring of $R$ is von Neumann regular.

A Lie algebra is semisimple if it has no nonzero solvable ideals, and an iUA-Lie ring is a Lie ring for which every commutator-preserving injection into another Lie ring is additiv…

Let $R$ be a Euclidean domain, and suppose that $R$ admits an ultra-Euclidean function. Ultra-Euclidean classification conjecture. Then either $R$ is a field or there is a field…

Let $R$ be a Euclidean domain, let $f:R\setminus\{0\}\to\mathbb{Z}_0^+$ be an ultra-Euclidean function, and let $\widetilde{f}$ denote the associated transformed function. Preserva…

Let $\mathbb{F}_q$ be a finite field of order $q=p^v$, where $p$ is prime and $v$ is a positive integer. For an integer $r>1$, call $a\in\mathbb{F}_q$ $r$-potent if $a^r=a$, and le…

Direct-product conjecture. The clean graphs satisfy

Extension-monad conjecture. The extension monad of a ring has the same underlying set as the extension monad of its additive reduct.

Let $\mathbb{F}$ be an arbitrary field, and let $\ell(\mathrm{M}_n(\mathbb{F}))$ denote the length of the full matrix algebra $\mathrm{M}_n(\mathbb{F})$. Paz's length conjecture. ……

Let $R$ be a ring. A ring is pseudo $\pi$-polar if it satisfies the pseudo $\pi$-polar condition, and $\pi$-UU if every unit $u\in U(R)$ has $1-u^n\in J(R)$ for some integer…

L'vov–Kaplansky conjecture. If $f$ is multilinear, then its image in the matrix algebra $M_n(K)$ is a vector space for every $n\in\mathbb{N}$.

Cossu–Zanardo necessity conjecture. If this matrix can be written as a product of two idempotent matrices, then it must satisfy the Cossu–Zanardo conjecture.

Cossu–Zanardo conjecture. There exist $a,b,c\in\mathbb{Z}[\sqrt{D}]$ with $a(1-a)=bc$ such that

Free-subgroup conjecture. Every non-central normal subgroup of $L_K(E)^\times$ contains a non-cyclic free subgroup.

Let $A$ and $B$ be rings and let $\mathcal{T}$ be a triangular ring satisfying the following conditions: for $a\in A$, $aA=0$ implies $a=0$; and for $b\in B$, $bB=0$ implies $b=0$.…

Existence conjecture. There is a weakly fadelian ring which is not fadelian.

A ring $R$ is nilperiod if every nilpotent element of $R$ is eventually annihilated by a power map in the sense established in the paper, and it is an NI-ring if its set of nilpote…

Let $\mathfrak{A}=\operatorname{Tri}(\mathcal{A},\mathcal{M},\mathcal{B})$ be a triangular algebra, and let $\mathcal{L}=\{L_n\}_{n\in\mathbb{N}}$ be a Lie triple higher derivation…

Let $\mathfrak{A}=\operatorname{Tri}(\mathcal{A},\mathcal{M},\mathcal{B})$ be a triangular algebra, and let $\mathcal{L}=\{L_n\}_{n\in\mathbb{N}}$ be a sequence of $\mathcal{R}$-li…

Let $R=\mathbb{Z}_{n}$, where … with primes $p_{1}<p_{2}<p_{3}<\cdots<p_{k}$, including $2$. Write $\psi(n)$ for the quantity appearing in the conjecture. Maximal-index conjecture.…

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

Let $(X,\leqslant)$ be a locally finite pre-ordered set, let $\mathcal{R}$ be a $2$-torsion-free commutative ring with unity, and let $I(X,\mathcal{R})$ denote the incidence algebr…

Let $m,n\geq 1$ be fixed integers, let $R$ be a semiprime ring satisfying suitable torsion restrictions, and let $T:R\longrightarrow R$ be a generalized $(m,n)$-Jordan centralizer.…

Let $m,n\geq 1$ be fixed integers, let $R$ be a semiprime ring satisfying suitable torsion restrictions, and let $F:R\longrightarrow R$ be a nonzero generalized $(m,n)$-Jordan deri…

BPS conjecture. begin{enumerate} item If $n>1$ is an odd integer, then