It is easy to see that the operation of ring multiplication preserves central idempotency: if are central idempotents, so is . In addition, if has a multiplicative identity , then is also a central idempotent. Furthermore, we may characterize central idempotency in a ring with as follows:
An idempotent in a ring with is central iff , where .
If is central, then clearly . Conversely, for any , we have . ∎
Another interesting fact about central idempotents in a ring with unity is the following:
The set of all central idempotents of a ring with has the structure of a Boolean ring.
First, note that . Next, for , we define addition and multiplication on as follows:
As discussed above, and are well-defined (as is closed under these operations). In addition, for any , we have
is a commutative monoid, in which every element is an idempotent (with respect to ). This fact is clear.
is commutative, since .
distributes over : we only need to show left distributivity (since is commutative by above):
This shows that is a Boolean ring. ∎
|Date of creation||2013-03-22 19:13:07|
|Last modified on||2013-03-22 19:13:07|
|Last modified by||CWoo (3771)|