Volume 1: Student Life Chapter 431: Questions Asked by Professor Ating
Seeing that Liu Maosheng hadn't said anything, Chen Zhou didn't ask any more questions.
On the podium, Professor Atin was talking incessantly about the theory of phased rings.
Graded ring theory is one of the important branches of ring theory, which refers to the theory of rings and modules with graded structures.
As for the research on graded rings and graded modules, it began as early as 1854.
At that time, Cayley introduced the group algebra K［G］ over the field K, which is the graded K-algebra of group G.
Another early example of a graded ring is the ring of polynomials over the real numbers R.
As Chen Zhou listened to Professor Ating's talk, he couldn't help but think of non-commutative rings.
Chen Zhou guessed that the reason why Professor Ating talked about graded ring theory was because he was looking for a breakthrough point in graded ring theory.
The main driving force for the initial development of graded rings and modules was projective algebraic varieties in commutative algebraic geometry, and they formed one of the basic methods in algebraic geometry research.
However, the reason why the development of graded rings and modules entered a new era was due to the promotion of non-commutative algebraic geometry and group representation theory.
The theory of group graded rings 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.
As more people study it, the development of this branch of mathematics is naturally promoted.
This is also the reason why any 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 the theory of group actions on rings and fixed points, in the theory of group representations, and especially in the theory of stable Clifford theory..."
After hearing what Professor Ating said, Chen Zhou became more certain of his guess.
The theory of graded rings is definitely the breakthrough point that Professor Artin is looking for.
At the podium, Professor Artin began to explain the role .
Under the podium , Chen Zhou began to multitask. On one hand, he was listening to Professor Ating's explanation, on the other hand, he was pondering the theory of fractional rings.
Chen Zhou had a fairly good understanding of the contents of the theory of successive rings.
After all, the information Professor Ating gave him contained some content on this aspect.
In addition to what Professor Artin just said, there is the theory of graded ordered groups of non-commutative rings, and the resulting theory of graded orders.
It is an important basic component of number theory, algebraic representation theory, non-commutative algebraic geometry, dimension theory and ring theory.
In addition, the theory of graded rings is very important.
However, what is more important is the research method of graded rings.
On the stage, Professor Artin has extended the theory to non-commutative rings.
Offstage, Chen Zhou was following the professor's ideas while thinking about the first property of graded ring theory.
This first property makes “A=⊕（n inN0）An=A0⊕A1⊕A2⊕…” a hierarchical ring.
Of course, this method of multitasking is mainly in line with Professor Ating’s thinking .
When it comes to so-called thinking, Chen Zhou only touches on it superficially and never goes deep into it.
Time passed quickly as Professor Atin spoke.
Chen Zhou also felt comfortable listening to it.
This kind of lecture, which is full of citations and completely independent of pre-prepared PPTs, sounds more interesting.
Of course, this also tests the professor's ability.
But this is not a problem at all for Professor Atin.
Because Chen Zhou has already discovered it.
Professor Atin’s PPT was a “prompter” from the very beginning.
This PPT has only 5 pages!
There are no more than 10 words on each page!
Basically, they are keywords used to hint at what is being said.
As for the specific content, it was all an impromptu speech developed by Professor Ating based on his own abilities.
"Now I finally understand." Liu Maosheng turned his head quietly and said something incoherent to Chen Zhou.
Chen Zhou asked in confusion: "What did you know?"
Liu Maosheng nudged Professor Ating on the podium with his mouth and said, "I finally understand why your mentor is so awesome!"
Zeng Zigu also came over and whispered, "The boss is the boss. I finally got to see it today."
Chen Zhou smiled softly: "Should I humbly accept your compliments on behalf of my mentor?"
Liu Maosheng immediately waved his hand and said with a chuckle, "That's not necessary. If there's a chance, you can bring Professor Ating with you and we can praise him in person."
Chen Zhou: “…”
Professor Ating on the podium turned to the last page of the PPT.
This page is about a generalization of a theorem in G-graded rings.
However, Professor Artin was not in a hurry to give his speech on the key words on this page.
Instead, he whispered a few words to the auditorium staff.
The staff understood what Professor Ating meant and left. Only then did Professor Ating start to return to the PPT.
"Here, we agree that G is a group, R is an associative ring with a unit element, and further assume that R is a G-graded ring, that is, R = ⊕ (g∈G) Rg..."
Chen Zhou was slightly stunned after listening to Professor Ating's words.
It turns out that this last page is not what Professor Ating himself wanted to talk about.
But it was prepared by Professor Ating for everyone.
That is, Professor Atin assigned an exercise.
It is a proof of the generalization of a theorem about G-graded rings.
On the stage, Professor Ating was still explaining the content of the "question".
Many students in the audience were dumbfounded.
Professor Artin, are you sure you are not mistaken?
Are you sure this is a proof of a generalization of the theorem?
Are you sure this is not an SCI proof?
They felt that Professor Ating was making things difficult for them.
For other professors, the exercises in class do have a certain limit.
You just come up with the research topic and prove it to be a paper.
Is this a bit too much?
Chen Zhou’s idea is actually similar.
But more people are still eager to try.
Chen Zhou felt that the question posed by Professor Ating was just right for testing his learning progress during this period.
Chen Zhouke did not slack off in studying the information he got from Professor Ating.
"…Then, we can get a corollary: Let M∈Mod(R▕S), then for A∈Zi∈S and Zi=⊕(g∈G)Zi, g{Re-modulo direct sum}."
"If g∈Supp(Zi), and there is a Re-isomorphic empty set: Zi, g→M, then the empty set uniquely expands to an R-isomorphic empty set e: Zi→M."
"As for this inference..."
As Ating spoke, he checked the time, then looked up at the people below the stage and said, "Give everyone 10 minutes to think about it."
After Professor Atin finished speaking, he walked down the podium and temporarily disappeared from everyone's sight.
Ating's words instantly made the audience below the podium noisy.
"Good fellow, you are indeed a master of mathematics. Who did you look down on during these 10 minutes?"
"Professor Ating, do you really think I can sort out the problem you just mentioned in 10 minutes?"
"Anyway, I can't figure it out. Look at my notes. I didn't remember everything..."
"I guess I can only look at what the professors say..."
