The intersection of any set of ideals of a ring is an ideal.

$R$ : ring, $D = \{I_1, \dots , I_n \}$ : arbitrary set of ideals of R.

Let's show that $S \subseteq D$, $\bigcap \limits_{I_{i} \in S}{I_{i}}$ is ideal of $R$.
Since that D satisfies $a - b \in I_{i}$ ($\forall a, \forall b \in \bigcap \limits_{I_{i} \in S}{I_{i}}$)($\forall I_{i} \in S$) and $ar, ra \in I_{i} \forall I_{i} \in S$.
Hence, by ideal test, it is trivial.