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?



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


