Primitive Idempotents and Constacyclic Codes over Finite Chain Rings

Let $R$ be a commutative local finite ring. In this paper, we construct the complete set of pairwise orthogonal primitive idempotents of $R[X]\mathrm{/}<g>$ where $g$ is a regular polynomial in $R[X]$. We use this set to decompose the ring $R[X]\mathrm{/}<g>$ and to give the structure of constacyclic codes over finite chain rings. This allows us to describe generators of the dual code ${\mathcal{C}}^{\mathrm{\perp }}$ of a constacyclic code $\mathcal{C}$ and to characterize non-trivial self-dual constacyclic codes over finite chain rings.