# Prime Avoidance Lemma for abelian groups?

I came across the following question

Let $$P_1, P_2$$, and $$P_3$$ be prime ideals of $$R$$. If an ideal $$I$$ satisfies $$I \subset P_1\cup P_2 \cup P_3$$, then prove $$I \subset P_i$$ for some $$i$$.

I proved it by doing many different cases and found out afterward that this a special case of the prime avoidance lemma (it is very long, looks correct to me). But, I am just curious, my solution never used the fact that we were working in ideals. In fact, it never used the fact that a ring is needed as I never multiplied anything.

So, I am wondering if this result holds for abelian groups? I assume not as I have not seen a result like this anywhere.

NOTE: After writing up my solution, I found a mistake. Comments below give a counterexample.

Thank You

• Klein four group is a union of three proper subgroups. Commented Jul 4, 2019 at 5:52

This is false for abelian groups, and even for ideals that are not required to be prime. For instance, let $$R=\mathbb{F}_2[x,y]/(x^2,xy,y^2)$$, let $$P_1=(x)$$, let $$P_2=(x+y)$$, and $$P_3=(y)$$. Then $$I=(x,y)$$ is equal to the union $$P_1\cup P_2\cup P_3$$, but is not contained in any of them. (If you have trouble verifying these claims, note that $$R$$ has only $$8$$ elements; its non-unit elements are just $$0,x,y,$$ and $$x+y$$.)

• That is weird, but I thought my proof worked for 3 abelian groups. Did I do something wrong?
– Mike
Commented Jul 2, 2019 at 20:18
• How are we supposed to know that without you showing us your proof?
– Con
Commented Jul 2, 2019 at 20:22
• @ThorWittich Right, let me write it up now
– Mike
Commented Jul 2, 2019 at 20:23
• The result does work for 2 abelian groups (and the usual proof of prime avoidance indeed does not use multiplication in that case). But for more than 2 a more complicated argument is needed that does not work without prime ideals. Commented Jul 2, 2019 at 20:24
• You should rather thank @EricWofsey. I only pointed out that we were not able to see your proof.
– Con
Commented Jul 2, 2019 at 20:51