8 problems

Let $R_{2n+1}$ denote the dihedral quandle of odd order $2n+1$, let $\Delta(R_{2n+1})$ be its fundamental ideal, and let $\Delta^{2n+1}(R_{2n+1})$ be its $(2n+1)$-st power. Let…

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 ${Xi}{i i I}$ 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 $X_1,X_2,\ldots,X_k$ be quandles, with associated cocycles $\phi_1,\ldots,\phi_k$ and cocycles $\psi_1,\ldots,\psi_k$ on the idempotent quandles…

Let $X$ be a quandle and let $\phi:X\times X\to A$ be a $2$-cocycle. Let … be the set of idempotents in the quandle ring $\mathbb{Z}_2[X]$, assume that $Y$ is a quandle, and let…

Let $X$ be a finite latin quandle, and let $\mathbb{Z}[X]$ denote its integral quandle ring. An idempotent is an element $u\in\mathbb{Z}[X]$ satisfying $u^2=u$; the trivial idempot…

Let $\operatorname{R}_n$ be the dihedral quandle and let $R$ be a ring. For the quandle ring $R[\operatorname{R}_n]$, let $\Delta(\operatorname{R}_n)$ denote its augmentation ideal…