Regular local ring implies integral domain
This article gives the statement and possibly, proof, of an implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property must also satisfy the second commutative unital ring property
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Statement
Any regular local ring is an integral domain. In other words, if a ring has a unique maximal ideal and that ideal is generated by a set whose size is the Krull dimension of the ring, then the ring is an integral domain.
Results used
Proof
Let be a regular local ring and
be its unique maximal ideal. We now prove the result by induction on the Krull dimension of
.
Base case for induction
If the dimension of is zero, then, by the definition of regular local ring, the maximal ideal must be trivial and hence, the ring must actually be a field, and hence an integral domain.
Induction step
Suppose the result is true for dimensions up to . We need to prove that the result is true for
of Krull dimension
.
We know the following:
- By Nakayama's lemma,
- The set of minimal prime ideals of
is finite (This needs to be clarified/explained properly)
Now, suppose were contained in the union of
and the minimal prime ideals. Then, by the prime avoidance lemma,
must be contained either in
or in one of the minimal prime ideals.
thus forces
to be a minimal prime ideal, which would make the Krull dimension zero, contradicting our assumption that the Krull dimension is at least 1.
Thus, there exists an element in
which is outside the union of
and all the minimal prime ideals. Let
and
. Clearly
is the unique maximal ideal in
. By the choice of
,
, and in fact we can conclude that
This needs to be clarified/explained properly.
Now is a proper homomorphic image of
so it can be generated by
elements. By Nakayama's lemma,
can also be generated by
elements.
Thus is a regular local ring, and hence, by the induction step assumption,
is an integral domain. Hence
is a prime ideal of
. But since
lies outside every minimal prime ideal, there is a minimal prime ideal properly contained inside
. Call this minimal prime ideal
.
If is any element, then we may write
for some
. But then since
,
. Thus,
. Nakayama's lemma now yields
, and hence
is an integral domain.