Chen Zhou was obviously stunned for a moment.

Is this a test for yourself as soon as you come up?

Geometrically Studying Non-Commutative Rings?

If you really want to talk about it, Chen Zhou still has some opinions about non-exchange rings.

Perhaps the most common example of a noncommutative ring is a matrix.

A number of counterexamples of noncommutative rings can be obtained by using the matrix.

For example, if S is a domain containing an infinite number of dimensions in the ring R.

Then A=Re_11+Re_12+Se_22, is Left Noether and Left Artin.

But not the right Noerther and the right Artin, which illustrates the difference between left and right chain conditions in non-commutative rings.

The finite direct product of all matrices on the ring division constitutes the so-called semi-unicyclic class.

This is commonly known as the Wedderburn-Artin theorem.

This is also the first brilliant structural theorem in noncommutative rings.

What's even more interesting is that it naturally illustrates the equivalence of the left semi-single ring to the right semi-single ring through the symmetrical structure of the matrix.

In swap rings, the two most common roots are the Jacobson root and the power zero root.

The former is simply called the Great Root, and it is the intersection of all the great ideals.

The latter is simply called the vegetarian root or small root, and it is the ideal cross of all vegetarians.

In the case of non-commutative, one root may be divided into three roots, satisfying certain conditions of left and right ideals and ideal intersections.

In fact, the intersection of all maximally left ideals of the noncommutative loop R is precisely the intersection of all maximally right ideals.

And they have inherited the corresponding reversible properties well.

Hence the Jacobson root of the non-commutative ring, also denoted as rad(R).

Although there is a distinction between left and right in non-commutative rings, there is also an interesting phenomenon of such divergent paths.

In commutative algebra, local loops have become a focus of research due to the widespread use of localization techniques.

However, the non-commutative loop technique seems to be limited.

On the contrary, I care about semi-local rings.

It is important to note that the definition of a semi-local ring in a non-commutative ring does not mean that it has only a finite number of maximal left ideals.

Rather, it is defined as whether R/rad(R) is a semi-unicycle or an Artin ring.

In fact, each (bilateral) ideal of the semi-local loop R contains rad(R), which can be reduced to the maximum ideal in the Artin ring R/rad(R), so there are at most a finite number of them.

But in the case of the left ideal, it is necessary to add the condition "R/rad(R) is commutative".

Otherwise, consider matrix algebra on the field, which is semi-local, but may have an infinite number of maximally left ideals.

As for the study of noncommutative rings from a geometric point of view, that is, the so-called method of studying commutative algebra from a local aspect.

The singularities in algebraic clusters are mainly discussed, and the properties of algebraic clusters around singularities.

But this is primarily for swapping rings, not non-swapping rings......

Chen Zhou's mind flashed through the content of the non-exchange ring.

However, this is only a half-baked understanding, and I have not studied it in depth.

Facing the mentor who met for the first time, he is still such a big guy.

What else can you see?

Instead of making an axe, he said some superficial understanding.

It's better to be honest and say that you don't have any opinions.

In front of such a math boss, he doesn't know how to pretend to understand, or deliberately pretends to be.

It's the really stupid thing.

Professor Atin saw that Chen Zhou had been silent and did not speak.

Then he smiled and asked, "What's the matter?" ”

Chen Zhou glanced at Professor Atin, and finally said honestly: "Professor, I don't have any opinion on studying noncommutative rings from a geometric perspective. ”

Hearing Chen Zhou's words, Professor Atin was stunned for a moment, but then relieved.

On the contrary, Chen Zhou's unbelieving approach left a good impression on him.

Laughing softly, Professor Atin said, "Yes, you are mainly studying analytic number theory. Maybe I should ask you, what do you think about the study of number theory?"

When Chen Zhou heard this, he also smiled.

It seems that Professor Atin is still quite easy to communicate.

Professor Atin looked at Chen Zhou and said, "The question just now is the content of my current research. ”

"As you know, my main area of research is algebraic geometry. As for number theory, maybe my father is more researched......"

When Professor Atin said this, there was obviously a hint of memory in his eyes.

He didn't shy away from this, but said with a smile: "When I'm older, I can't help but miss the past." ”

Chen Zhou smiled kindly and expressed understanding.

Professor Atin then continued: "So, after you enter the university, you can join me in studying algebraic geometry, or you can delve into the problems of number theory yourself. ”

"I don't have any restrictions on that. Of course, as your mentor, if you have any questions, you can come to me. I'll do my best to answer for you. ”

Regarding Professor Ting's words, Chen Zhou still had some expectations.

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.

Human time is limited, and human energy is limited.

How to give full play to the limited energy in the limited time is the most important thing.

In this regard, Chen Zhou naturally has his own ideas.

So, he replied, "Thank you, Professor Artin, for your understanding and for your honesty. ”

Professor Artin: "So, what are you going to do?"

Chen Zhou: "I want to follow you to learn the content of algebraic geometry on the one hand, and on the other hand, not to lose the study of analytic number theory. ”

Chen Zhou chose the most greedy choice.

That is, to grasp with both hands, and to ask for 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 that.

At least, Chen Zhou knew what he wanted and had his own plan.

This is much better than the PhD students he has taken in the past.

They only know how to complete the tasks they have assigned.

So, Atin said: "In terms of time, I will not limit you. I believe that as a brilliant young mathematician, you can organize your time. ”

"But first you have to go back and think about how to study noncommutative rings geometrically. ”

After a pause, Atin added with a smile: "In addition, you shouldn't have to worry about your graduation thesis, right?"

When Chen Zhou heard this, he immediately smiled and said, "Professor, don't worry. ”

When he left Professor Artin's office, Chen Zhou left with a thick pile of printer paper.

The printed content is also all about Professor Ting's research materials.

Chen Zhou raised his hand and looked at his watch, it was 10 o'clock in the morning.

It should be too late to go to Professor Friedman's again.

I just don't know what kind of content this Nobel laureate in physics, Professor Jerm Friedman, will arrange for himself.

For the study and research of physics, Chen Zhou prefers to follow his mentor.

Chen Zhou took out the beginner's manual, flipped to the page of the map, and searched for it.

He locked Professor Friedman's office.