In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, the prime avoidance lemma says that if an ideal ''I'' in a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''R'' is contained in a
union of finitely many
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s ''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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
that a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces.
Statement and proof
The following statement and argument are perhaps the most standard.
Statement: Let ''E'' be a subset of ''R'' that is an additive subgroup of ''R'' and is multiplicatively closed. Let
be ideals such that
are prime ideals for
. If ''E'' is not contained in any of
's, then ''E'' is not contained in the union
.
Proof by induction on ''n'': The idea is to find an element that is in ''E'' and not in any of
's. The basic case ''n'' = 1 is trivial. Next suppose ''n'' ≥ 2. For each ''i'', choose
:
where the set on the right is nonempty by inductive hypothesis. We can assume
for all ''i''; otherwise, some
avoids all the
's and we are done. Put
:
.
Then ''z'' is in ''E'' but not in any of
's. Indeed, if ''z'' is in
for some
, then
is in
, a contradiction. Suppose ''z'' is in
. Then
is in
. If ''n'' is 2, we are done. If ''n'' > 2, then, since
is a prime ideal, some
is in
, a contradiction.
E. Davis' prime avoidance
There is the following variant of prime avoidance due to E. Davis.
Proof:
[Adapted from the solution to ] We argue by induction on ''r''. Without loss of generality, we can assume there is no inclusion relation between the
's; since otherwise we can use the inductive hypothesis.
Also, if
for each ''i'', then we are done; thus, without loss of generality, we can assume
. By inductive hypothesis, we find a ''y'' in ''J'' such that
. If
is not in
, we are done. Otherwise, note that
(since
) and since
is a prime ideal, we have:
:
.
Hence, we can choose
in
that is not in
. Then, since
, the element
has the required property.
Application
Let ''A'' be a Noetherian ring, ''I'' an ideal generated by ''n'' elements and ''M'' a finite ''A''-module such that
. Also, let
= the maximal length of ''M''-
regular sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection.
Definitions
Fo ...
s in ''I'' = the length of ''every'' maximal ''M''-
regular sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection.
Definitions
Fo ...
in ''I''. Then
; this estimate can be shown using the above prime avoidance as follows. We argue by induction on ''n''. Let
be the set of associated primes of ''M''. If
, then
for each ''i''. If
, then, by prime avoidance, we can choose
:
for some
in
such that
= the set of zerodivisors on ''M''. Now,
is an ideal of
generated by
elements and so, by inductive hypothesis,
. The claim now follows.
Notes
References
*
Mel HochsterDimension 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
Algebra