# Prime avoidance lemma of E. Davis [Exercise 16.8 in Matsumura, Commutative Ring Theory, related with finite union of prime ideals]

Let $$A$$ be a commutative unitary ring, $$P_1,\dots , P_r$$ prime ideals, $$I$$ an ideal, and $$x$$ an element of $$A$$. If $$xA+I\not \subset P_1\cup \cdots \cup P_r$$, then there is $$y\in I$$ such that $$x+y\not \in P_1\cup \cdots \cup P_r$$.

I have tried several methods but nothing works. For example, I tried to apply the following theorem:

Let $$\mathfrak{a}$$ be an ideal, $$P_1,\dots ,P_r$$ prime ideals. If $$\mathfrak{a}\subset P_1\cup \cdots \cup P_r$$, there exists $$1\leq k \leq r$$ such that $$\mathfrak{a}\subset P_k$$.

And of course, I tried to show contraposition but failed. If you know how to solve, please help me!!

Matsumura attributes this to E. Davis, and the only paper of Davis in Matsumura's References section is "Ideals of the principal class, $$R$$-sequences and a certain monoidal transformation" (link). The following argument is from the proof of the Lemma in the Appendix.

We may assume that there are no containments between the $$P_{i}$$, i.e. $$P_{i_{1}} \not\subseteq P_{i_{2}}$$ if $$i_{1} \ne i_{2}$$.

By assumption, there exist $$a \in A$$ and $$u \in I$$ such that $$xa+u \not\in P_{i}$$ for all $$i$$. This implies that if $$x \in P_{i}$$ then $$u \not\in P_{i}$$. Set $$S := \{i \in \{1,\dotsc,r\} \;:\; x \in P_{i}\}$$ and $$S' := \{1,\dotsc,r\} \setminus S$$.

If $$S'$$ is empty, then we take $$y := u$$. Suppose $$S'$$ is nonempty; then by prime avoidance, the set $$\bigcap_{i \in S'} P_{i} \setminus \bigcup_{i \in S} P_{i}$$ is nonempty; let $$b$$ be an element of this set; we take $$y := bu$$.

If $$i \in S$$, then $$x \in P_{i}$$ implies $$u \not\in P_{i}$$; since $$b \not\in P_{i}$$, we have $$x+bu \not\in P_{i}$$.

If $$i \in S'$$, then $$x \not\in P_{i}$$ but $$b \in P_{i}$$ so $$x+bu \not\in P_{i}$$.

• What if $S$ is empty and $\bigcap_{i \in S;}P_{i}=0$? May 16, 2021 at 19:54
• @user124697 yes, then we can take $b = 0$ so that $y = 0$. May 16, 2021 at 21:49