Counterexamples to the avoidance lemma for arbitrary ideals

Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following:

(1) If $J_k$, $k=1,\dots,n$, are prime ideals, then there exists some $j$ such that $I\subseteq J_j$.

(2) If $A$ is a principal ideal ring, then there exists some $j$ such that $I\subseteq J_j$.

I want some counterexamples that for all $k=1,\dots,n$, $I$ is not contained in $J_k$. How to get the counterexamples?

• I suggest you to read this paper. – user89712 Nov 7 '13 at 12:34

As a simple source of counterexamples, let $A$ be any finite ring that is not a principal ideal ring. Then for any nonprincipal ideal $I\subset A$, $I$ is the union of the (finitely many) principal ideals generated by the elements of $I$, but it is not contained in any single such principal ideal. A concrete example of such a ring is $A=k[x,y]/(x^2,xy,y^2)$ for any finite field $k$, with $I=(x,y)$.