Definition

For a ring , an orthogonal family of central idempotents is a subset such that

Decomposing rings

Using the same argument as used for central idempotents, we can take any element to give

This gives the decomposition

Let . Let be a th root of unity (for example, we can take ). For , let

Then by computation,

Looking specifically at the term in parentheses,

If then the sum is , which gives

If then is a non-trivial th root of unity and

so .

Finally, the coefficient on is

If , then we have the coefficient is . If then this sum is

Thus,

so this satisfies all the conditions of an orthogonal family of central idempotents.

Therefore,