An example ellucidating proposition 2.8 in Milne's notes on commutative algebra

Question

I would like to know if one can weaken conditions of Proposition 2.8 in http://www.jmilne.org/math/xnotes/CA.pdf

The proposition says that if an ideal $a$ in a ring $A$ is contained in the union of ideals $p_1,...,p_r$ with $p_2,...,p_r$ prime, then $a$ is contained in one of $p_i$.

Why do we need to require that $p_2,...,p_r$ are all prime? What would be the simplest example where some of $p_i$ are non-prime and the proposition does not hold?

It seems to me at least that if $A$ is a polynomial ring $\mathbb C[x_1,...,x_n]$ then one does not need to require that $p_i$ are prime. Am I wrong?

Answer 1

Some variants of prime avoidance (as this property is usually called) are in Eisenbud's "Commutative Algebra with a View..." on p. 114, including an example where it fails: for instance the ideal $(x,y)$ in $\mathbb Z/2\mathbb Z[x,y]/(x,y)^2$ is the union of three (smaller) non-prime ideals.

Answer 2

For an infinite base field $k$, you are right. Assume $a\subset\cup_{i=1}^r a_i$ and choose $r$ the least such. Then we can pick $f_i\in a$ not in the union of the the $a_j$'s with $j\neq i$. Consider the $k$-vector space $V$ generated by the $f_i$'s and let $V_i=a_i\cap V$. Then $V\subset \cup V_i$, but $V_i$s are proper subspaces of $V$, a contradiction, since $k$ is infinite.