In ring theory, a branch of mathematics, a Peirce decomposition /ˈpɜːrs/ is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
A Peirce decomposition for Jordan algebras (which are non-associative) was introduced by Albert (1947).
If e is an idempotent element (e2 = e) of an associative algebra A, the two-sided Peirce decomposition of A given the single idempotent e is the direct sum of eAe, eA(1 − e), (1 − e)Ae, and (1 − e)A(1 − e).
In general, given idempotent elements e1, ..., en which are mutually orthogonal and sum to 1, then a two-sided Peirce decomposition of A with respect to e1, ..., en is the direct sum of the spaces ei A ej for 1 ≤ i, j ≤ n. The left decomposition is the direct sum of ei A for 1 ≤ i ≤ n and the right decomposition is the direct sum of Aei for 1 ≤ i ≤ n. Generally, given a set e1, ..., em of mutually orthogonal idempotents of A which sum to esum rather than to 1, then the element 1 − esum will be idempotent and orthogonal to all of e1, ..., em, and the set e1, ..., em, 1 − esum will have the property that it now sums to 1, and so relabeling the new set of elements such that n = m + 1, en = 1 − esum makes it a suitable set for two-sided, right, and left Peirce decompositions of A using the definitions in the last paragraph.
This is the generalization of the simple single-idempotent case in the first paragraph of this section.