An inclusion relation between ideals in a commutative ring

Question

Let $\{I_i\}$ be a family of ideals in a commutative ring $R$, and let $R[x]$ be the polynomial ring over $R$. We know that if for an ideal $J$ of $R$, $J[x]\subseteq\cup I_i[x] $, then $J \subseteq\cup I_i $ (where $J[x]$ is the extension of $J$ in $R[x]$). Is the converse statement true, that is, if $J \subseteq\cup I_i$ is it true that $J[x]\subseteq\cup I_i[x]$?

Answer 1

It is not true.

For a counterexample, let $A$ be the Klein four-group (i.e., the abelian group $\left(\mathbb Z/2\mathbb Z\right)^2$), written additively. Define a ring $R$ by $R = \mathbb Z \oplus A$ as additive groups, with multiplication being defined by

$\left(n,a\right)\left(m,b\right) = \left(nm,nb+ma\right)$.

(This ring structure is a particular case of what is called a Dorroh extension.) Regard $A$ as a subset of $R$ via the canonical injection $A \to \mathbb Z \oplus A = R$. Then, $A$ is an ideal of $R$. Let $u$, $v$ and $w$ be the three nonzero elements of $A$. Then, $A \subseteq uR \cup vR \cup wR$, but $A\left[x\right] \not\subseteq uR\left[x\right] \cup vR\left[x\right] \cup wR\left[x\right]$ (for instance, the polynomial $u + vx$ lies in the left hand side, but not in the right).