Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t]-submodules Im(1 − t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Zn[t ]/(t − a) where gcd(n, a) = 1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. ...

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is generalized here to arbitrary monomial ideals. It is shown how this duality is naturally expressed by Bass numbers, in their relations to the Betti numbers ...