  Let $K$ be a finite extension of $\mathbb{Q}$ and $\mathcal{O}_K$ be its ring
of integers. Let $\mathfrak{B}$ be a primitive collection of ideals in
$\mathcal{O}_K$. We show that any $\mathfrak{B}$-free system is essentially
minimal. Moreoever, the $\mathfrak{B}$-free system is minimal if and only if
the characteristic function of $\mathfrak{B}$-free numbers is a Toeplitz
sequence. Equivalently, there are no ideal $\mathfrak{d}$ and no infinite
pairwise coprime collection of ideals $\mathcal{C}$ such that
$\mathfrak{d}\mathcal{C}\subseteq\mathfrak{B}$. Moreover, we find a periodic
structure in the Toeplitz case. Last but not least, we describe the
restrictions on the cosets of ideals contained in unions of ideals.
