Cyclic codes over the ring \(\mathbb {Z}_{2^{k}} + u\mathbb {Z}_{2^{k}}\)

The purpose of this manuscript is two-fold. First, properties of the ring \(\mathcal {R}_{k} = \mathbb Z_{2^{k}} + u\mathbb Z_{2^{k}}\) ${\mathcal{R}}_k={\mathbb{Z}}_{2^k}+u{\mathbb{Z}}_{2^k}$ and the set of ideals are established. Second, results on cyclic codes of length n, \(\gcd (2,n)=1\) $gcd(2,n)=1$ , over the non-chain Frobenius ring \(\mathcal {R}_{k}\) ${\mathcal{R}}_k$ and their description by means of idempotent elements are presented.