Cohen–Macaulay modules over some non-reduced curve singularities

In this article, we study Cohen–Macaulay modules over non-reduced curve singularities. We prove that the rings k〚x,y,z〛/(xy,yq−z2)k〚x,y,z〛/(xy,yq−z2) have tame Cohen–Macaulay representation type. For the singularity k〚x,y,z〛/(xy,z2)k〚x,y,z〛/(xy,z2) we give an explicit description of all indecomposable Cohen–Macaulay modules and apply the obtained classification to construct families of indecomposable matrix factorizations of x2y2∈k〚x,y〛x2y2∈k〚x,y〛.