$I,I_1,...,I_n$ be ideals in a commutative ring with unity such that $I \subseteq \cup_{j=1}^n I_j$ , then is $I\subseteq I_k$ for some $k$?

Question

Let $I,I_1,...,I_n$ be ideals in a commutative ring with unity such that $I \subseteq \cup_{j=1}^n I_j$ , then is it true that $I\subseteq I_k$ for some $k$ ? I know the result is true for $n=2$ , and for general $n$ given $I_1,..,I_n$ are prime ideals . But I cannot prove or disprove it for general $n$ for arbitrary ideals . Please help . Thanks in advance

Answer 1

I think this is the standard counterexample. The ring is $\mathbb Z_2 [x,y]$. Then consider these ideals: $$ I_1 = ( x, y^2), \ \ \ I_2 = ( y, x^2), \ \ \ I_3 = ( x + y, xy ), \ \ \ \ \ \ \ \ \ I = ( x , y ).$$