Xueba starts with change

Chapter 404 The Most Greedy Choice
Chen Zhou was obviously taken aback.

Is it just to test yourself as soon as you come up?

Studying non-commutative rings geometrically?
To be honest, Chen Zhou still has some opinions on non-commutative rings.

Perhaps the most common example of a non-commutative ring is a matrix.

A number of counterexamples of non-commutative rings can be obtained by using matrices.

It is as if, if S is the field of corresponding infinite dimension contained in the ring R.

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

But not right Noerther and right Artin, which shows that the chain condition has left and right differences in non-commutative rings.

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

This is commonly known as the Wedderburn-Artin theorem.

This is also the first wonderful structure theorem in noncommutative rings.

What's more interesting is that it naturally shows that the left semi-monocyclic is equivalent to the right semi-monocyclic 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 Dagen, and it is the intersection of all great ideals.

The latter is called prime root or small root for short, and it is the cross of all prime ideals.

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

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 well inherit the corresponding reversible properties.

Therefore, it is called the Jacobson root of the noncommutative ring, also denoted as rad(R).

Although there is a distinction between left and right in non-commutative rings, there are also such interesting phenomena that lead to the same goal through different routes.

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

However, the local ring technology of non-commutative rings seems to be limited.

On the contrary, it is particularly concerned about semi-local rings.

It is worth noting that the definition of a semilocal ring in a noncommutative ring does not mean that it has only a finite number of maximal left ideals.

Instead, it is defined that R/rad(R) is a semi-single ring or an Artin ring.

In fact, each (two-sided) ideal of the semilocal ring 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.

But for the left ideal situation, it is necessary to add the condition "R/rad(R) can be exchanged".

Otherwise, matrix algebra over fields can be considered, which is semi-local, but there may be infinitely many maximal left ideals.

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

The singular points in algebraic varieties and the properties of algebraic varieties around singular points are mainly discussed.

But this is mostly for commutative rings, not non-commutative rings...

Chen Zhou's mind quickly flashed the content about the non-commutative ring.

However, this is only a half-baked understanding, and I haven't studied it in depth.

Facing the mentor whom he met for the first time, he was still such a big boss.

How else can I see it?
Instead of playing tricks, talk about some simple understanding.

It's better to be honest and say that I have no opinion.

In front of such a mathematician, pretend to understand, or deliberately show off.

That's the really stupid thing to do.

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

Then he smiled and asked again: "What's the matter? If you have any ideas, you can just say it."

Chen Zhou glanced at Professor A Ting, and finally said honestly: "Professor, I have no opinion on the study of non-commutative rings from a geometric point of view."

Hearing Chen Zhou's words, Professor Arting was stunned for a moment, but he was relieved immediately.

On the contrary, Chen Zhou's unreliable behavior left a good impression on him.

With a soft smile, Professor Artin said, "That's right, you are mainly studying analytic number theory. Maybe I should ask you, what do you think about number theory research?"

Chen Zhou also smiled when he heard the words.

It seems that Professor Arting is quite easy to communicate with.

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

"You also know that my main research field is algebraic geometry. As for number theory, maybe my father is more researched..."

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

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

Chen Zhou smiled kindly, expressing his understanding.

Then Professor Artin continued: "So, after you enroll, you can join me to study algebraic geometry, and you can also delve into the problems of number theory by yourself."

"I have no restrictions on this. Of course, as your mentor, you can come to me if you have any questions. I will try my best to answer them for you."

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

After all, no mentor can ignore his current achievements in the field of analytic number theory.

Not to mention, forcing him to change his research direction.

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.

Chen Zhou naturally had his own thoughts on 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 follow you to learn the content of algebraic geometry, and on the other hand, not to lose the research of analytic number theory."

Chen Zhou chose the greediest choice.

That is to say, grasping with both hands and wanting with both hands.

Professor Arting was stunned for a moment, but quickly realized.

He understood what Chen Zhou meant.

And he didn't think there was anything wrong with it.

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

This is much better than those doctoral students he has supervised in the past.

They only know how to complete the tasks they assign.

So, Artin said: "In terms of time, I will not limit you. I believe that as an outstanding young mathematician, you can arrange your own time."

"However, you have to go back and think about how to study non-commutative rings geometrically."

After a pause, Arting added with a smile: "Besides, you don't need to worry about your graduation thesis, do you?"

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

When leaving Professor A Ting's office, Chen Zhou left with a stack of thick printing paper.

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

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

I will go to Professor Friedman again, there should be still time.

I just don’t know what kind of content the Nobel Prize winner in physics, Professor Jerome Friedman, will arrange for himself.

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

Chen Zhou took out the novice manual, turned to the page of the map, and searched for it.

They locked Professor Friedman's office.

(End of this chapter)