  We give a formula of the connected component decomposition of the Alexander
quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots,
f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots,
f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic
to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see
how a quandle is decomposed into connected components for some Alexander
quandles. We introduce a decomposition of a quandle into the disjoint union of
maximal connected subquandles. In some cases, this decomposition is obtained by
iterating a connected component decomposition. We also discuss the maximal
connected sub-multiple conjugation quandle decomposition.
