In ring theory, a branch of mathematics, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements. The Peirce decomposition for associative algebras was introduced by . A Peirce decomposition for Jordan algebras (which are non-associative) was introduced by .
If e is an idempotent element (e<sup>2</sup> = 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 â e), (1 â e)Ae, and (1 â e)A(1 â e). There are also corresponding left and right Peirce decompositions. The left Peirce decomposition of A is the direct sum of eA and (1 â e)A and the right decomposition of A is the direct sum of Ae and A(1 â e).
In those simple cases, 1 â e is also idempotent and is orthogonal to e (that is, e(1 â e) = (1 â e)e = 0), and the sum of 1 â e and e is 1. In general, given idempotent elements e<sub>1</sub>, ..., e<sub>n</sub> which are mutually orthogonal and sum to 1, then a two-sided Peirce decomposition of A with respect to e<sub>1</sub>, ..., e<sub>n</sub> is the direct sum of the spaces e<sub>i</sub> Ae<sub>j</sub> for 1 ⤠i, j ⤠n. The left decomposition is the direct sum of e<sub>i</sub> A for 1 ⤠i ⤠n and the right decomposition is the direct sum of Ae<sub>i</sub> for 1 ⤠i ⤠n.
Generally, given a set e<sub>1</sub>, ..., e<sub>m</sub> of mutually orthogonal idempotents of A which sum to e<sub>sum</sub> rather than to 1, then the element 1 â e<sub>sum</sub> will be idempotent and orthogonal to all of e<sub>1</sub>, ..., e<sub>m</sub>, and the set e<sub>1</sub>, ..., e<sub>m</sub>, 1 â e<sub>sum</sub> will have the property that it now sums to 1, and so relabeling the new set of elements such that n = m + 1, e<sub>n</sub> = 1 â e<sub>sum</sub> 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.
An idempotent of a ring is called central if it commutes with all elements of the ring.
Two idempotents e, f are called orthogonal if ef = fe = 0.
An idempotent is called primitive if it is nonzero and cannot be written as the sum of two nonzero orthogonal idempotents.
An idempotent e is called a block or centrally primitive if it is nonzero and central and cannot be written as the sum of two orthogonal nonzero central idempotents. In this case the ideal eR is also sometimes called a block.
If the identity 1 of a ring R can be written as the sum
of orthogonal nonzero centrally primitive idempotents, then these idempotents are unique up to order and are called the blocks or the ring R. In this case the ring R can be written as a direct sum
of indecomposable rings, which are sometimes also called the blocks of R.