Throughout is a ring.
Theorem (Jacobson). If for every there exists some such that then is commutative.
The proof of Jacobson’s theorem can be found in standard ring theory textbooks. Here we only discuss a very special case of the theorem, i.e. when for all
Definitions. An element is called idempotent if The center of is
It is easy to see that is a subring of An element is called central if Obviously is commutative iff i.e. every element of is central.
Problem. Prove that if for all then is commutative.
Solution. Clearly is reduced, i.e. has no nonzero nilpotent element. For every we have and so is idempotent for all Hence, by Remark 3 in this post, is central for all Now, since
we have and thus is central. Also, since
we have and hence is central. Therefore is central.
A similar argument shows that if for all then is commutative (see here!).