Give an example of a commutative ring with unit and an ideal that has no primary decomposition.
I think boolean Ring will be the right example, but I don't know how I must show that. So please help me.
Boolean ring is $R=P(\mathbb{N})$ (the power set of $\mathbb{N}$) with $$A+B=(A\cup B)-(A\cap B)$$ $$AB=A\cap B$$
Also if you have another example, please tell me, thank you.