Schur rings over infinite dihedral group

Schur rings over the infinite dihedral group $\mathcal{Z}⋊{\mathcal{Z}}_2$ are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over $\mathcal{Z}⋊{\mathcal{Z}}_2$ are classified under the assumption that $\mathcal{Z}$ is an $\mathcal{A}$-subgroup. Those Schur rings are proved to be traditional.