For some ring $R$ (no $1$ or commutativity necessary) and $a \in R$, we defined the principal ideal $(a)$ by
$$ (a) := \bigcap \{ I : a \in I \subseteq R, I \text{ is an ideal}\}.$$
Now as a homework question, we shall show that $(a)$ always is of the form $$ (a) = \{ra + as + z\cdot a + \sum_{i=1}^m r_ias_i : r,a,r_i,s_i \in R, z\in \mathbb Z \},$$
where $z\cdot a$ is defined as repeated addition of $a$ with itself. Of course, any element of this form is contained in $(a)$, but I'm clueless as how to show the opposite inclusion. Could you give any hints on how to arrive at that any $x\in (a)$ can be written in this special sum form?