When doing my 3rd-year project, I found I needed a classical theorem in commutative algebra called Krull’s Intersection Theorem. It turned out to be very useful; I used it in three proofs involving formal power series, and it also gave me some intuition for DVRs. However, despite being an intuitive result, the proofs I found in Zariski–Samuel and Atiyah–MacDonald required quite a bit more commutative algebra knowledge than I currently possess. Thankfully, a much simpler proof was found in 2004 by Hervé Perdry. Here is a presentation of that proof, which I originally found in Milne’s *Primer* and to whom I give the credit. I post it here simply to make this useful theorem more widely accessible.

**Theorem**

Let be an ideal in the Jacobson radical (ie. the intersection of all maximal ideals) in a Noetherian ring . Then

As most sources are quick to point out, this “intuitive” fact can fail if the ring is not Noetherian.

**Proof**

We first wish to prove that

The inclusion is obvious. For the reverse inclusion, let generate , and let be the set of all homogeneous polynomials in letters over of degree such that for , we have that . Generate an ideal from , and choose a finite generating subset .

Let be the maximum degree of all the ‘s, and let . Then can be represented as where is a homogeneous polynomial over of degree . Therefore

Since and each are homogeneous, we can take ‘s to be homogeneous too, and of degree , since any terms not of this degree will cancel anyway and so are redundant. It follows that no is constant since was specifically chosen to be higher degree than any . Thus

We have that and . Therefore .

Now use Nakayama’s lemma to deduce the result.

Original source: http://www.jmilne.org/math/xnotes/ca.html

Next semester I enter my final year. This week, I selected the courses I’ll be taking.

**Masters project:**Not quite sure what this will involve yet. The supervisor says he will like me to be introduced to sheaf cohomology but hasn’t decided where to take that.**Algebraic Geometry:**Natch. The syllabus changed a lot this year because there is a new lecturer taking it, but it looks great.**Manifolds, Homology and Morse Theory:**Interesting looking course with one of my favourite lecturers.**Number Theory****Higher Arithmetic:**When I mention this one people think I’m taking a class in multiplying very large numbers.**Measure Theory:**I felt I could not leave my undergraduate without learning about measure and Lebesgue integration.**Complex Dynamical Systems:**My university has a dynamical systems research group and the lecturer is very passionate about the subject, so this should be good.

A bit of a resolution is to take better advantage of office hours. I’m one of those students who maybe for reasons of vanity (fear of looking stupid, for example; totally irrational, I know) doesn’t make much use of office hours, but not taking advantage of access to professional mathematicians is a great way to hobble oneself.

Leave a comment about what you’ll be doing next year!