Definition: An element in a ring is called central if is in the center of i.e. for all
1) Suppose is a ring in which Let be an idempotent, i.e. Prove that is central.
Hint:
Spoiler:
2) Use 1) to give a short proof of this very special case of Jacobson's theorem: if for all in a ring with identity then is commutative.
Hint:Spoiler:
3) This time suppose for all in a ring with identity Use 1) to prove that is commutative.
Hint:Spoiler:
4) Challenge: Suppose for all in a ring with identity Use 1) to prove that is commutative.
Okay, let’s try again.
and are central, so and are central. So is central. Moreover
is central. Hence is central. Hence is central, and so is central. Hence is central. Now
is also central, and so is central, and so is central. Replacing by gives that is central. Hence is central.
At last! Hence is central.
My question is related to:
2) Use 1) to give a short proof of this very special case of Jacobson's theorem: if for all in a ring with identity then is commutative.
Hint: Spoiler:
clearly is idempotent, and thus central by 1), for any Show that and are also central.
Let center of the ring be C.
Clearly C is a sub-ring of R
Also
To prove
Consider
From the closure property of sub-ring we get
Is my approach correct?
I am having trouble in proving . Any push plz?
Thanks. It look so easy when you see it
I'm now stuck with
3) This time suppose for all in a ring with identity Use 1) to prove that is commutative.
Hint: Spoiler:
It's immediate that is an idempotent, and thus central by 1), for all Put to show that is central for all
As suggested in the hint - I could prove the following:
1.
2.
3.
How do I show R is commutative? A push again plz?
Infact I was trying to show that ab+ba=0. Then using (1) above I will show
ab = ba and hence will be done. But no success.
Thanks. And now for the last part.
4) Challenge: Suppose for all in a ring with identity Use 1) to prove that is commutative.
I completely follow TheAbstractionist proof. I have just two question.
1. I would never have been able to do this proof. Is there a structured line of attack which you were following here? Or this just a matter of lot of practice and off-course lot of gray-matter?
2. Do we end at 5? Or this is true in general, where ?
the idea is to look for idempotents.
the result is true for all the interesting thing is that can even change as the element changes. so, the Jacobson's theorem says:2. Do we end at 5? Or this is true in general, where ?
if is a ring with identity such that for every there exists an integer such that then is commutative.
of course, understanding the proof of this theorem requires a fairly deep knowledge of ring theory. there are, however, special cases of
this theorem which can be proved quite easily. for example if is finite or, more genrally, Artinian, then the theorem is just a quick result
of Artin-Wedderburn theorem.