Volume 1: Student Life Chapter 404: The Greediest Choice
Chen Zhou was obviously stunned for a moment.
Are you going to test yourself right from the start?
Study non-commutative rings from a geometric perspective?
To be honest, Chen Zhou does have some opinions about non-commutative rings.
Perhaps the most common example of a noncommutative ring is a matrix.
Using matrices, we can obtain a number of counterexamples of non-commutative rings.
It is like, if S is a field whose corresponding dimension is infinite contained in the ring R.
Then A=Re_11＋Re_12＋Se_22, which is left Noether and left Artin.
But it is not right Noerther and right Artin, which shows that the chain conditions have left and right differences in non-commutative rings.
The finite direct product of all matrices over the division ring forms the class of so-called semisimple rings.
This is commonly known as the Wedderburn-Artin theorem.
This is also the first wonderful structural theorem in non-commutative rings.
What’s more interesting is that it naturally shows that the left semi-simple ring is equivalent to the right semi-simple ring through the symmetric structure of the matrix.
In commutative rings, the two most common roots are the Jacobson root and the nilpotent root.
The former is simply called the big root, which is the intersection of all great ideals.
The latter is simply called the prime root or small root, which is the intersection of all prime ideals.
In the non-commutative case, a root may be split into three roots that satisfy certain conditions of left and right ideals and ideal intersections.
In fact, for the non-commutative ring R, the intersection of all maximal left ideals is exactly the intersection of all maximal right ideals.
And they inherit the corresponding reversible properties well.
Therefore it is called the Jacobson root of a noncommutative ring, also denoted by rad(R).
Although there is a distinction between left and right in non-commutative rings, there are also many interesting phenomena that lead to the same result in different ways.
In commutative algebra, local rings have become a research focus due to the widespread use of localization techniques.
But the local ring technique for non-commutative rings seems to be limited.
On the contrary, I am particularly concerned about semi-local rings.
It is worth noting that the definition of a semilocal ring in a non-commutative ring does not mean that it has only a finite number of maximal left ideals.
Instead, it is defined as R/rad(R) is a semi-simple ring or an Artin ring.
In fact, every (bilateral) ideal of a semilocal ring R contains rad(R) and can be reduced to a maximal ideal in the Artin ring R/rad(R), so there are at most finitely many of them.
But for the left ideal case, the condition "R/rad(R) is commutative" must be added.
Otherwise, one can consider matrix algebra over the field, which is semilocal but may have infinitely many maximal left ideals.
As for studying non-commutative rings from a geometric point of view, that is, the so-called method of studying commutative algebra from a local aspect.
This paper mainly discusses the singular points in algebraic varieties and the properties of algebraic varieties around the singular points.
But this is mainly for commutative rings, not non-commutative rings...
Content about non-commutative rings flashed rapidly through Chen Zhou's mind.
However, I only have a half-understood understanding and have not studied it in depth.
The mentor I met for the first time turned out to be such a big shot.
How else can I see it?
Instead of showing off my knowledge in front of experts, I would rather talk about some simple understandings.
It’s better to just say honestly that I don’t have any opinion .
In front of such a math giant, pretend to know everything or show off deliberately.
That's the really stupid thing.
Professor Ating saw that Chen Zhou remained silent and did not say anything.
Then he smiled and asked, "What's wrong? If you have any ideas, just tell me."
Chen Zhou glanced at Professor Ating and finally said honestly: "Professor, I have no opinion on studying non-commutative rings from a geometric perspective."
After hearing Chen Zhou's words, Professor Ating was stunned for a moment, but then he was relieved.
On the contrary, Chen Zhou's approach of not speaking without thinking left a good impression on him.
Professor Atin chuckled and said, "That's right. You mainly study analytic number theory. Perhaps I should ask you what you think about number theory research?"
Chen Zhou also smiled when he heard this.
It seems that Professor Ating is quite easy to communicate with.
Professor Ating looked at Chen Zhou and said, "The question just now is the content of my current research."
"As you know, my main research area is algebraic geometry. As for number theory, perhaps my father is more knowledgeable..."
When Professor Atin said this, there was obviously a hint of nostalgia in his eyes.
He did not shy away from talking about this, but smiled and said, "As I get older, I can't help but miss the past."
Chen Zhou smiled kindly to show his understanding.
Then Professor Atin continued, "So, after you enroll, you can join me in studying algebraic geometry, or you can delve into number theory problems on your own."
"I don't set any limits on this. Of course, as your mentor, if you have any questions, you can come to me. I will do my best to answer them for you."
Chen Zhou had some expectations about what Professor Ating would say.
After all, with his current achievements in the field of analytic number theory, no mentor can ignore it.
Not to mention, it forced him to change the direction of his research.
People’s time is limited, and people’s energy is limited.
The most important thing is how to make full use of limited energy in limited time.
Chen Zhou naturally has his own ideas about this.
So, he replied: “Professor Artin, thank you for your understanding and frankness.”
Professor Artin: "So, what are you going to do?"
Chen Zhou: "I want to learn algebraic geometry from you while not giving up the study of analytic number theory."
Chen Zhou chose the most greedy option.
That is to say, we need to do both.
Professor Atin was stunned for a moment, but quickly reacted.
He understood what Chen Zhou meant.
And he didn't think there was anything wrong with it.
At least, Chen Zhou knows what he wants and has his own plan.
This is much better than the doctoral students he has supervised in the past.
They only know how to complete the tasks assigned to them.
So, Artin said, "I will not restrict you in terms of time. I believe that as an outstanding young mathematician, you can manage your time well."
"But first you have to go back and think carefully about how to study non-commutative rings from a geometric perspective."
After a pause, Ating smiled and added, "Besides, I don't need to worry about your graduation thesis, right?"
Upon hearing this, Chen Zhou immediately smiled and said, "Professor, don't worry."
When Chen Zhou left Professor Ating's office, he left with a thick stack of printed paper.
The printed content is all about Professor Ating’s research materials.
Chen Zhou raised his hand and looked at his watch. It was 10 o'clock in the morning.
It should still be in time to go to Professor Friedman now.
I just don't know what kind of content Professor Jerome Friedman, the Nobel Prize winner in physics, will arrange for himself.
When it comes to studying and researching physics, Chen Zhou prefers to follow his tutor.
Chen Zhou took out the beginner's manual, turned to the map page, and searched for a while.
Then they locked onto Professor Friedman's office.
