Commutative Algebra - Paul Nelson - 2018

Charlotte, 23.01. 8:00-8:30

1. A commutative ring, x in A st for all primes P there is some n st x^n is in P. What can you say about x?

2. The following questions from the problem sheet: 3. 9. and 7.

Andreas, 23.01. 11:00-11:30

We started with a little bit of small talk about the exercises in general and moved to the same question as stated above: A commutative ring, x in A st for all primes P there is some n st x^n is in P. What can you say about x? The solution is that x is nilpotent, use that the intersection of all primes is the nilradical. Then he asked me how to prove this and I started to give the proof from the first chapter in Atiyah-MacDonald. He interrupted me quite fast, after I established the setting, and he asked for a different proof using localization. But here I was not sure and finished the proof of A-M orally. Then we discussed about the exercises 3,9 and 7 from the list, Good luck!

Anonymous, 24.01. 11:00-11:30

Prof. Nelson started it off by asking what I thought about the questions on the handout and whether I was able to do them. I said I was able to solve most of them except for parts of 7,9, and 10.

1. The first question given was same as the first one mentioned above: suppose in a ring A there is an element x such that for all primes there exists some power of x that is in the prime, what can you say about x? The answer I gave was that x should be nilpotent. I mentioned that one can probably prove this directly, but I gave a proof using localization:

Suppose by way of contradiction, that x is not nilpotent, then any power of x is nonzero. Therefore the localization A[1/x] is nonzero. Take a maximal ideal in A[1/x], its preimage in A is a prime ideal such that no power of x is contained in it. Thus we have a contradiction.

2. Following the previous question, he asked: Give an example of an integral domain A, and an element x in A such that x is non-nilpotent (I thought about this later that actually one cannot have a nilpotent element in an integral domain), and x is contained in every non-zero prime ideal of A. I tried some stuff that didn't work, then he hinted that I should think of a local ring of dimension one, in which case any element in the maximal ideal will work. So I gave the example of the ring of formal power series in one variable. Another student has told me he asked this same question in their exam as well; it seems that he just came up with this question today as a follow-up to the first question, so be prepared.

3. He asked me to do problems 4, 9, and 5 from the handout.

Benjamin, 25.01. 08:00-08:30

Same introductory questions + jacobson radical in polynomial rings

We then discussed questions 4,7,8,9; he usually stopped me very quickly if I seemed certain how to proceed. (While doing exercise 8 he interrupted me after I wrote the free resolution down, asked me what my result was and then moved on to a question about whether/ for which i Tor_i(M,N) vanishes, given that these are both f.g. modules over some ring->pid->Z).

Raphael, 25.01. 08:30-09:00

At the start he asked me about the Problem sheet, and if I had any problems with any of the exercises.

1. The first two questions were the same as above + the jacobson radical of k(x)

2. 8,9,1a (just the statement),7a

Felix, 25.01. 10:00-10:30

He asked the questions about the nilradical, the Jacobson radical of a polynomial ring and Tor of f.g. Z modules.

Then we discussed 7,8,9. Afterwards he asked if A is in B and B is Noetherian, is A noetherian.