Consider $R = \bigoplus_{i=1}^{n} \mathbb{Z}/k\mathbb{Z}$ for some $n\in\mathbb{N}$ and some composite $k \in\mathbb{N}$. Clearly it forms a module over $\mathbb{Z}/k\mathbb{Z}$.
I want to count maximal cardinality of a subset $B \subset R$ such that for every $b \in B$ there exists some $i \in\{1, 2, \ldots , k-1\}$ such that $i.r \not\in B$. I worked through some examples to possibly arrive at a pattern, but have not succeeded yet. I would appreciate any help with this.
If $k=p$ is prime then any largest subset $B_p$ will not contain $0$ and for each line $\in (\Bbb{F}_p{}^n -0)/\Bbb{F}_p^\times$ exactly one element will be removed, which means that $$|B_p| = p^n-1-\frac{p^n-1}{p-1}$$
If $k=p^e$ is a prime power then an optimal subset will be $$B_{p^e}=(\Bbb{Z}/p^e \Bbb{Z})^n - (p^{e-1} \Bbb{F}_p{}^n - p^{e-1} B_p), \qquad |B_{p^e}|=p^{en}-(1+\frac{p^n-1}{p-1})$$
If $k=\prod_{j=1}^J p_j^{e_j}$ then I'm not sure. Maybe $$B_k=((\Bbb{Z}/p_j^{e_j} \Bbb{Z})^n-0)\oplus (\Bbb{Z}/\frac{k}{p_j^{e_j}} \Bbb{Z})^n$$ where $p_j^{e_j}$ is the largest prime power dividing $k$.