Give a finite abelian group $$G \cong \mathbb{Z}_{m_1} \oplus \cdots \oplus \mathbb{Z}_{m_{t}}=\langle(1,0, \ldots 0), (0,1, \ldots, 0), \ldots, (0,0, \ldots, 1)\rangle ,$$ where $m_i \mid m_{i+1}$ for $i=1, \ldots, t-1$.
Suppose we have two cyclic subgroups $H_1 = \langle(a_1,a_2, \ldots, a_t)\rangle,$ and $H_2 = \langle(b_1,b_2, \ldots, b_t)\rangle,$ is there a way to tell that $H_1$ and $H_2$ have trivial intersection from $a_i$'s and $b_i$'s?
Clearly, $H_1$ and $H_2$ have nontrivial intersection iff there exists distinct $k,l \in \{1, \ldots, m_t-1\}$, such that $ka_i \equiv lb_i \pmod {m_i}$ for all $i$. I was wondering if there is another equivalent and cleaner criteria.
Any reference would be really appreciated.