Prime avoidance lemma

{{Short description|Result concerning ideals of commutative rings}} {{Expand French|Lemme d'évitement des idéaux premiers|date=October 2018|topic=sci}}

In algebra, the '''prime avoidance lemma''' says that if an ideal ''I'' in a commutative ring ''R'' is contained in a union of finitely many prime ideals ''P''_''i'''s, then it is contained in ''P''_''i'' for some ''i''.

There are many variations of the lemma (cf. Hochster); for example, if the ring ''R'' contains an infinite field or a finite field of sufficiently large cardinality, then the statement follows from a fact in linear algebra that a vector space over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces.<ref>Proof of the fact: suppose the vector space is a finite union of proper subspaces. Consider a finite product of linear functionals, each of which vanishes on a proper subspace that appears in the union; then it is a nonzero polynomial vanishing identically, a contradiction.</ref>

== Statement and proof == The following statement and argument are perhaps the most standard.

'''Theorem''' (''Prime Avoidance Lemma'')''':''' Let ''E'' be a subset of commutative ring ''R'' that is an additive subgroup of ''R'' and is multiplicatively closed. (In particular, ''E'' could be a subring or ideal of ''R''.) Let <math>I_1, I_2, \dots, I_n, n \ge 1</math> be ideals such that <math>I_i</math> are prime ideals for <math>i \ge 3</math>. If ''E'' is not contained in any of the <math>I_i</math>, then ''E'' is not contained in the union <math display="inline">\bigcup I_i</math>.

'''Proof by induction on ''n'':''' The idea is to find an element of ''R'' that is in ''E'' and not in any of the <math>I_i</math>. The base case <math>n=1</math> is trivial. Next suppose <math>n\geq 2</math>. For each ''i'', choose :<math>z_i \in E \setminus \bigcup_{j \ne i} I_j</math>, where each of the sets on the right is nonempty by the inductive hypothesis. We can assume <math>z_i \in I_i</math> for all ''i''; otherwise, there is some <math>z_k</math> among them that avoids all of the <math>I_i</math>, and we are done. Put :<math>z = z_1 \cdots z_{n-1} + z_n</math>. Because ''E'' is closed under addition and multiplication, ''z'' is in ''E'' by construction. We claim that ''z'' is not in any of the <math>I_i</math>. Indeed, if <math>z\in I_i</math> for some <math>i \le n - 1</math>, then <math>z_n\in I_i</math>, a contradiction. Next suppose <math>z\in I_n</math>. Then <math>z_1 \cdots z_{n-1} \in I_n</math>. If <math>n=2</math>, this is already a contradiction. If <math>n>2</math>, then, since <math>I_n</math> is a prime ideal, <math>z_i\in I_n</math> for some <math>i\leq n-1</math>, again a contradiction. <math>\square</math>

== E. Davis' prime avoidance == There is the following variant of prime avoidance due to [https://www.genealogy.math.ndsu.nodak.edu/id.php?id=5703 E. Davis].

{{math_theorem|math_statement=<ref>{{harvnb|Matsumura|1986|loc=Exercise 16.8.}}</ref> Let ''A'' be a ring, <math>\mathfrak{p}_1, \dots, \mathfrak{p}_r</math> prime ideals, ''x'' an element of ''A'' and ''J'' an ideal. For the ideal <math>I = xA + J</math>, if <math>I \not\subset \mathfrak{p}_i</math> for each ''i'', then there exists some ''y'' in ''J'' such that <math>x + y \not\in \mathfrak{p}_i</math> for each ''i''.}}

'''Proof:'''<ref>Adapted from the solution to {{harvnb|Matsumura|1986|loc=Exercise 1.6.}}</ref> We argue by induction on ''r''. Without loss of generality, we can assume there is no inclusion relation between the <math>\mathfrak{p}_i</math>'s; since otherwise we can use the inductive hypothesis.

Also, if <math>x \not\in \mathfrak{p}_i</math> for each ''i'', then we are done; thus, without loss of generality, we can assume <math>x \in \mathfrak{p}_r</math>. By inductive hypothesis, we find a ''y'' in ''J'' such that <math>x + y \in I - \cup_1^{r-1} \mathfrak{p}_i</math>. If <math>x + y</math> is not in <math>\mathfrak{p}_r</math>, we are done. Otherwise, note that <math>J \not\subset \mathfrak{p}_r</math> (since <math>x \in \mathfrak{p}_r</math>) and since <math>\mathfrak{p}_r</math> is a prime ideal, we have: :<math>\mathfrak{p}_r \not\supset J \, \mathfrak{p}_1 \cdots \mathfrak{p}_{r-1}</math>. Hence, we can choose <math>y'</math> in <math>J \, \mathfrak{p}_1 \cdots \mathfrak{p}_{r-1}</math> that is not in <math>\mathfrak{p}_r</math>. Then, since <math>x + y \in \mathfrak{p}_r</math>, the element <math> x + y + y'</math> has the required property. <math>\square</math>

=== Application === Let ''A'' be a Noetherian ring, ''I'' an ideal generated by ''n'' elements and ''M'' a finite ''A''-module such that <math>IM \ne M</math>. Also, let <math>d = \operatorname{depth}_A(I, M)</math> = the maximal length of ''M''-regular sequences in ''I'' = the length of ''every'' maximal ''M''-regular sequence in ''I''. Then <math>d \le n</math>; this estimate can be shown using the above prime avoidance as follows. We argue by induction on ''n''. Let <math>\{ \mathfrak{p}_1, \dots, \mathfrak{p}_r \}</math> be the set of associated primes of ''M''. If <math>d > 0</math>, then <math>I \not\subset \mathfrak{p}_i</math> for each ''i''. If <math>I = (y_1, \dots, y_n)</math>, then, by prime avoidance, we can choose :<math>x_1 = y_1 + \sum_{i = 2}^n a_i y_i</math> for some <math>a_i</math> in <math>A</math> such that <math>x_1 \not\in \cup_1^r \mathfrak{p}_i</math> = the set of zero divisors on ''M''. Now, <math>I/(x_1)</math> is an ideal of <math>A/(x_1)</math> generated by <math>n - 1</math> elements and so, by inductive hypothesis, <math>\operatorname{depth}_{A/(x_1)}(I/(x_1), M/x_1M) \le n - 1</math>. The claim now follows.

== Notes == {{reflist}}

== References == *Mel Hochster, [http://www.math.lsa.umich.edu/~hochster/615W10/supDim.pdf Dimension theory and systems of parameters], a supplementary note *{{cite book |last1 = Matsumura |first1 = Hideyuki |year = 1986 |title = Commutative ring theory |series = Cambridge Studies in Advanced Mathematics |volume = 8 |url = {{google books|yJwNrABugDEC|Commutative ring theory|plainurl=yes|page=123}} |publisher = Cambridge University Press |isbn = 0-521-36764-6 |mr = 0879273 |zbl = 0603.13001 }}

Category:Abstract algebra