Xueba starts with change

Chapter 431 Questions from Professor Artin (Supplement)

Seeing that Liu Maosheng hadn't spoken, Chen Zhou didn't ask any more questions.

On the podium, Professor Arting was eloquent on the content of fractionated ring theory.

Graded ring theory is one of the important branches of ring theory, which refers to the theory of rings and modules with a graded structure.

As for the research on graded rings and graded modules, it started as early as 1854.

At that time, Cayley introduced the group algebra K[G] over the field K, which is a graded K-algebra of the group G.

Another early example of a graded ring is a polynomial ring over the field R of the real numbers.

Listening to Professor A Ting's narration, Chen Zhou couldn't help but think of non-commutative rings.

Chen Zhou reckoned that the reason why Professor A Ting talked about the hierarchical ring theory was also because he was looking for a breakthrough point from the hierarchical ring theory.

The main impetus for the initial development of graded rings and modules is to exchange projective algebraic varieties in algebraic geometry, and form one of the basic methods in the study of algebraic geometry.

However, the reason for the development of graded rings and modules to enter a new era is the promotion of non-commutative algebraic geometry and group representation theory.

Group graded ring theory is very active and fruitful.

Also because of its close connection with many branches of mathematics, group graded rings have aroused the interest of a large number of mathematicians.

And when there are more people doing research, the development of this branch of mathematics will naturally be promoted.

This is also the reason why the branch of mathematics, or any field, can continue to develop.

"An example of graded ring theory is the theory of arbitrary group grades of noncommutative rings, which plays an important role in group actions on rings and fixed points, group representation theory, and especially the stable Clifford theory... ..."

Hearing Professor A Ting's words, Chen Zhou became more determined in his guess.

The graded ring theory is definitely a breakthrough point that Professor Arting is looking for.

On the podium, Professor Artin began to explain the role of graded ring theory in terms of Clifford theory.

Under the podium, Chen Zhou began to focus on two tasks. While listening to Professor A Ting's explanation, he was thinking about the hierarchical ring theory by himself.

Chen Zhou still understands the content of the fractional ring theory.

After all, Professor Arting gave him some information about this.

In addition to what Professor Artin just said, the theory of the ordered group of non-commutative rings, and the resulting theory of the order of the order.

It is an important building block of number theory, algebraic representation theory, noncommutative algebraic geometry, dimension theory, and ring theory.

Also, the theory of fractional rings, though, is important.

However, more important is the research method of graded rings.

On the stage, Professor Artin has extended to the non-commutative ring.

In the audience, Chen Zhou not only followed the train of thought of the professor on the stage, but also thought about the first attribute of the graded ring theory.

The first attribute is to make "A=⊕(n inN0)An=A0⊕A1⊕A2⊕..." a hierarchical ring.

Of course, this kind of dual-purpose approach mainly followed the train of thought of Professor A Ting.

The so-called thinking, Chen Zhou just tasted it, never went deep.

With Professor Artin's narration, time passed quickly.

Chen Zhou was also very comfortable listening.

This kind of quotes from many sources, completely separated from the pre-prepared PPT lectures, sounds more interesting.

Of course, this also tests the professor's ability even more.

But this is not a problem at all for Professor Artin.

Because, Chen Zhou has already found out.

Professor Artin's PPT has been a "teleprompter" from the very beginning.

This PPT is only 5 pages in total!
There are no more than 10 words on each page!
Basically, keywords are used to suggest what is being said.

As for the specific content, it was all an unscripted speech delivered by Professor Arting with his own ability.

"I finally know now." Liu Maosheng tilted his head quietly, and said something endless to Chen Zhou.

Chen Zhou asked puzzledly, "What do you know?"

Liu Maosheng nuzzled Professor A Ting on the podium with his mouth: "I finally know why your tutor is so awesome!"

Zeng Zigu also came over and said in a low voice: "The boss is the boss, I finally saw it today."

Chen Zhou smiled softly: "Should I accept your praise humbly on behalf of my mentor?"

Liu Maosheng waved his hand immediately, and said with a smile: "That's no need, if you have a chance, you take Professor A Ting with you, and we will praise him face to face."

Chen Zhou: "..."

Professor Arting on the podium turned the PPT to the last page.

This page is about a generalization of a theorem in G-graded rings.

However, Professor Artin was not in a hurry to give his own speech on the key words on this page.

Instead, he whispered a few words to the staff in the auditorium.

The staff understood what Professor A Ting meant, and after leaving, Professor A Ting started to return to the PPT.

"Here, we agree that G is a group, R is an associative ring with identity elements, and further assume that R is a G-graded ring, that is, R=⊕(g∈G)Rg..."

Listening to Professor A Ting's words, Chen Zhou was slightly taken aback.

It turned out that the last page was not what Professor Arting himself was going to talk about.

It was prepared by Professor Artin for everyone.

That is, Professor Artin assigned an exercise.

It is a proof of a generalization of a theorem about G-graded rings.

On the stage, Professor A Ting was still talking about the content of the "question".

In the audience, many students began to be dumbfounded.

Professor Artin, are you sure you are not mistaken?

Are you sure this is a proof of theorem generalization on the fly?

Are you sure this is not an SCI proof?

They felt that Professor Arting was embarrassing.

For other professors, there is still a limit to the exercises in class.

When you come out with that research topic directly, it proves to be a thesis.

Is this a little too much?
Chen Zhou's thoughts are actually similar.

But more, but still eager to try.

Chen Zhou felt that the question posed by Professor A Ting was just to test his study during this period.

Chen Zhou did not slack off on the materials he got from Professor A Ting.

"...Then, we can get a corollary: Let M∈Mod(R▕S), then for ∈Zi∈S and Zi=⊕(g∈G)Zi, g{Re-Modular Direct Sum}."

"If g∈Supp(Zi), and there is a Re-homomorphism: Zi, g→M, then the unique expansion becomes R-homomorphism e: Zi→M."

"As for this inference..."

As Ah Ting spoke, he looked at the time, then raised his head and said to the crowd below the podium: "Just give everyone 10 minutes to think about it."

After Professor A Ting finished speaking, he stepped off the podium and temporarily disappeared from everyone's sight.

And what Arting said instantly made the audience under the podium become noisy.

"Good guy, you are indeed a master of mathematics, who are you looking down on in these 10 minutes?"

"Professor A Ting, do you really think I can figure out what you just said in 10 minutes?"

"Anyway, I don't understand it. Hey, look at my notes. I haven't memorized them all..."

"I guess, I can only look at those professors..."

(End of this chapter)