Unsolved Problems

A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal....

Can every non-negative rational function be expressed as a sum of squares of rational functions?...

If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f...

If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?...

Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?...

Are there infinitely many Leinster groups?...

Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal?...

Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?...

Conjecture There exists a finite lattice which is not the congruence lattice of a finite algebra....