commutativity theorems on rings
Since Wedderburn proved his celebrated theorem that any finite division ring is commutative, the interest in studying properties on a ring that would render the ring commutative dramatically increased. Below is a list of some of the so-called “commutativity theorems” on a ring, showing how much one can generalize the result that Wedderburn first obtained. In the list below, is assumed to be unital ring.
In each of the cases below, is commutative:
(Wedderburn’s theorem) is a finite division ring.
(Jacobson) If for every element , there is a positive integer (depending on ), such that .
(Jacobson-Herstein) For every , if there is a positive integer (depending on ) such that
(Herstein) If there is an integer such that for every element such that , the center of .
(Herstein) If for every , there is a polynomial ( depending on ) such that .
(Herstein) If for every , such that there is an integer (depending on ) with
Some of the commutativity problems can be derived fairly easily, such as the following examples:
If is a ring with such that for all , then is commutative.
Let . From the assumption, we have . Expanding the LHS, we get . Expanding the RHS, we get . Equating both sides and eliminating common terms, we have
Similarly, from , we expand the equations and get
Finally, expanding out and eliminating common terms, keeping in mind Equations (1) and (2) from above, we get . ∎
If each element of a ring is idempotent, then is commutative.
If contains , then we can apply Theorem 2: for for any . Otherwise, we do the following trick: first , so that for all . Next, , so , which implies , and the result follows.
The corollary also follows directly from part 2 of Theorem 1. ∎
|Title||commutativity theorems on rings|
|Date of creation||2013-03-22 17:54:55|
|Last modified on||2013-03-22 17:54:55|
|Last modified by||CWoo (3771)|