25 problems

Let $G$ be a torsion-free group, let $\mathbb{F}$ be a field, and let $\mathbb{F}[G]$ be the group ring. The obvious units in $\mathbb{F}[G]$ are the elements $\lambda g$ with…

Let $\Gamma$ be a torsion-free group, and let $\mathbf{C}\Gamma$ be its group algebra over the complex numbers. A zero divisor is a nonzero element that annihilates another nonzero…

Let $G$ be a group, let $\mathbb C G$ be its complex group algebra, and let $\ell^2(G)$ be the Hilbert-space completion of $\mathbb C G$ in the 2-norm. Linnell's analytic zero divi…

Zero-Divisor Conjecture. The ring $\mathbb C\Gamma$ contains no zero-divisors.

Let $G$ be a torsion-free group, and let $\mathbb{Q}[G]$ denote its rational group ring. Zero-divisor conjecture. The ring $\mathbb{Q}[G]$ has no non-trivial zero divisors. This co…

Uniqueness conjecture. In $U_q^w$ and $U_q^v$, the only zero divisors annihilating all generators are $J_w-1$ and $J_v-1$, respectively.

Let $X$ be a finite-dimensional non-degenerate evolution ${\mathbb K}$-algebra, and write its optimal direct-sum decomposition into indecomposable evolution ideals as … For an evol…

For a prime $p$, let $q_p(2)$ denote the Fermat quotient of $2$, and let $\log_{\mathcal{A}}(2)$ be its associated element of $\mathcal{A}$. A Wieferich prime is a prime satisfying…

Positive-signature variant conjecture. The split algebra ${\mathbb A}_{3,1}$ involves variants of ${\mathbb P}_{4}$ with positive signature.

Let $R$ be a standard graded algebra, and let $d>0$. A pair $(\ell,Q)$ of exact zero divisors means that $\ell$ is a general linear form, $Q$ is an element of degree $d-1$, and the…

Let $R=bigoplus{n i G}Rn$ be a ring graded by a torsion-free group $G$, and let $I$ be a left ideal of $R$. The ideal $I$ is unfaithful when its annihilator is nonzero. **The g…

Let $R$ be a ring, let $G$ be a torsion-free group, and let $R[G]$ be the group-ring. An element is a zero-divisor if it annihilates a nonzero element. The generalized zero-divisor…

Let $\varGamma$ be a group and let $\Bbbk\varGamma$ be its group algebra over the field $\Bbbk$. The group $\varGamma$ is torsion-free if it has no nonidentity element of finite or…

Pattern-preserving weak equivalence conjecture. The embedding $\mathcal I$ is a weak homotopy equivalence.

Let $G$ be a group and let $\mathbb C G$ be its complex group algebra. An element $\alpha\in\mathbb C G$ is regular if $\alpha\beta\neq 0$ for every nonzero $\beta\in\mathbb C G$.…

Let $R$ be a Noetherian local ring, let $M$ be a nonzero $R$-module of finite type and finite projective dimension, and let $r\in R$ be not a zero-divisor on $M$. Auslander's zero-…

Let $\Bbbk$ be a field of characteristic zero and let $G$ be a torsion-free group. An idempotent in the group ring $\Bbbk[G]$ is an element $p$ satisfying $p^2=p$. The trivial idem…

Let $R$ be a Noetherian local ring, let $M$ be an $R$-module of finite type and finite projective dimension, and let $r\in R$ be not a zero-divisor on $M$. Auslander's zero-divisor…

Let $\Gamma$ be a torsionfree amenable group, let $p$ be a prime, and let $\mathcal M$ be a left $\mathbb Z\Gamma$-module satisfying … The preceding corollary asserts that…

Let $\Gamma$ be a torsion-free group and let $\mathbb{C}\Gamma$ denote its group algebra. A zero divisor is a nonzero element of an algebra that annihilates a nonzero element. The…

Primitivity conjecture. Every pair of zero-divisors in $\mathbb Z[G]$ is primitive.

Let $G$ be a torsion-free group and let $R$ be an integral domain. A non-trivial zero divisor in the group ring $R[G]$ is a nonzero element $\alpha\in R[G]$ for which there exists…

Let $K$ be a division ring and let $G$ be a group. The Zero Divisors Conjecture. If $K[G]$ contains zero divisors, then $G$ has torsion. The source states that the Invertibles Conj…

Linnell's conjecture. The convolution $f*g$ is nonzero.

Zero divisor conjecture. The group ring ${\mathbb Z}H$ has no nontrivial right zero divisors.