12 problems

Let $X$ be a finite latin quandle, and let $u\in\mathbb{Z}[X]$ be a nonzero idempotent, so $u^2=u$. An element of $X$ is called a trivial idempotent, and $\varepsilon$ denotes the…

Let $\mathcal{E}$ be a complex evolution algebra. An element $u\in\mathcal{E}$ is an idempotent if $u^2=u$. If $\mathcal{E}$ has natural basis $\{e_1,\dots,e_n\}$ and structure mat…

Let $R$ be a field and let $G$ be a group. Kaplansky's idempotent conjecture. If $G$ is torsionfree, then the group ring $RG$ has only the trivial idempotents, namely $0$ and $1$.…

Let ${Xi}{i i I}$ be a family of quandles, and suppose that each integral quandle ring $\mathbb{Z}[Xi]$ has only trivial idempotents. Let $\star{i i I}Xi$ denote their free product q…

For a quandle $X$, its integral quandle ring is $\mathbb{Z}[X]$, and an idempotent is an element $e$ satisfying $e^2=e$. The idempotents $0$ and the basis elements of $X$ are calle…

Let $R=\bigoplus\limits_{n\in G}R_n$ be a ring graded by an Abelian group $G$, and let $H$ be the torsion subgroup of $G$. An idempotent is an element $e$ satisfying $e^2=e$. The s…

Let $R=\bigoplus\limits_{n\in G}R_n$ be a ring graded by a torsion-free group $G$, where $0$ denotes the identity element of $G$. An idempotent is an element $e$ satisfying $e^2=e$…

The idempotent criterion conjecture. A subspace $M\subseteq V$ is a Mathieu–Zhao subspace of $V$ if and only if, whenever $e_i\in M$, one has $V^i\subseteq M$. This is proposed as…

Let $X$ be a quandle and let $\mathbb{Z}[X]$ denote its integral quandle ring. A quandle is semi-latin when every right multiplication restricted to each orbit is injective; a fini…

Let $A$ be a generic algebra, and let $\Pi$ be a $k$-dimensional affine subspace of $A$. An idempotent is an element $e\in A$ satisfying $e^2=e$. The idempotent bound. The affine s…

Idempotent conjecture. If $e\in\operatorname{Im}\delta$ is an idempotent, then

Let $L$ be a meet semi-lattice with a fixed linear extension, and let $D$ range over the valid diagrams of $L$, meaning subsets containing the minimal elements and stable under joi…