If is an idempotent in a ring (not necessarily with identity), then we can decompose as the direct sum of four components (subrings), each of them related to . Concretely,
note that here is just notation that means , and similarly for .
This decomposition is known as the Peirce decomposition of with respect to .
An important feature of this decomposition is that each of and is a subring of , and the former even has an identity, namely, the element , which does indeed lie in , since ; also, for all .
If does contain an identity , then is defined, and is again an idempotent, and one has (obviously), and since as is idempotent. One says that and form a pair of orthogonal idempotents. Furthermore, in this case, the subring as defined above, coincides with the set of products . (A similar remark applies to each of and .)
More generally, continuing to assume that contains an identity, if is a set of orthogonal idempotents for with the property that then
The basic idea of decomposing a ring via idempotents should be accessible to anyone knowing the basics of ring theory. Several of the examples below refer to contexts which require a more specialized degree of knowledge; in such cases, this is indicated at the beginning of the example.
If lies in the centre of (in particular, if is commutative), then , and Thus we obtain the simpler decomposition
This example requires a basic knowledge of the costruction of the spectrum of a commutative ring with identity.
If is commutative with and is an idempotent, then , as noted in Example 1. As remarked above, we may form the idempotent , and so in particular, both and are commutative rings with identity (namely and respectively). Thus , and are all defined, and the factorization induces a decomposition
of into a union of two open subsets. Conversely, any such decomposition of arises from an idempotent in this way.
In short, if we think of as being the ring of regular functions on , then idempotents in serve precisely as the indicator (or characteristic) functions of simultaneously open and closed subsets of . (In particular, is the indicator function of itself, while is the indicator function of the empty subset.)
Note that in the case that is a Boolean ring, so that every element is idempotent (by definition), the space is totally disconnected, and the above discussion specializes to the Stone representation theorem.
Suppose that is a topological group which admits a neighbourhood basis of the identity consisting of compact open subgroups. (A basic example of such a group is , the general linear group of invertible -matrices over the field of -adic numbers.) One consequence of this assumption is that is locally compact, and so we can choose a Haar measure on .
Let denote the -vector space of all compactly supported locally constant -valued functions on . We can make into a -algebra by defining a convolution product on its elements as follows:
for any two functions . The -algebra is then referred to as the Hecke algebraof the group .
If is any compact open subgroup of , then we may define an idempotent via the formula where is the indicator (or characteristic) function of . Since is open, the function is locally constant and compactly supported, and thus lies in , and hence so does .
In this situation, for any , the convolution is the function obtained by averaging on the left via the action of the compact open subgroup , and simililarly is the is the function obtained by averaging via the action of on the right. In particular if and only if for all and , and simililarly, if and only if for all K and . Thus the subalgebra of consists of those elements of that are bi-invariant under the action of , i.e. such that for all and .
Since any is compactly supported and locally constant, by virtue of our assumption on we may find some sufficiently small compact open subgroup such that is bi--invariant. Thus we find that
where the union is indexed by the collection of all compact open subgroups of .