93 problems

Let $R$ be a finite chromatic ring, and let $\Gamma(R)$ be its zero-divisor graph. Write $\omega(\Gamma(R))$ for the clique number of $\Gamma(R)$ and $\chi(\Gamma(R))$ for its chro…

Let $R$ be a ring, and let $\operatorname{Nil}^*(R)$ denote its upper nilradical, the largest nil ideal of $R$. A one-sided ideal $J$ of $R$ is called nil if every element of $J$ i…

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.

Salce–Zanardo conjecture. If $R$ satisfies $\operatorname{ID}_2$, then $R$ must be a Bézout domain.

Faith's conjecture. Every left or right perfect, left self-injective ring is quasi-Frobenius.

A ring $R$ is strongly right Johns if every homomorphic image of $R$ is right Johns; a ring is right Johns when its right annihilator structure satisfies the Johns condition. Faith…

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.…

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

A ring is called $*$-regular if it is regular and its involution is proper. It is directly finite if $ab=1$ implies $ba=1$, and unit-regular if for every element $a$ there is a uni…

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

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

Vukman–Kosi-Ulbl conjecture. The map $d$ must be a derivation whose image is contained in the center $Z(R)$.

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

Ring-homomorphism conjecture. A ring homomorphism generally corresponds to integrable or partially solvable systems.

Let $S$ be a commutative ring and let $\mathbb{M}_2(S)$ denote the ring of $2\times 2$ matrices over $S$. An element is left strongly regular if it satisfies $a=a^2x$ for some $x$…

Guo's conjecture. Every $R$-module has an $(R,S)$-injective envelope.

Let $R$ be a ring. The claim concerns the properties of being strongly weakly nil-clean, being semi-potent, and being a WUU ring. Strongly weakly nil-clean characterization. A ring…

A ring $R$ is uniquely weakly nil-clean if it is weakly nil-clean and every nil-clean element is uniquely nil-clean. WUU characterization. A ring $R$ is a WUU ring if, and only if,…

Let $\mathfrak{R}$ be a ring admitting an automorphism of prime period $p$ such that every fixed point of the automorphism lies in the center of $\mathfrak{R}$. Herstein's conjectu…

Ideal-quotient conjecture. Under these hypotheses, $R$ is Armendariz of the Hurwitz series type.