Peirce decomposition

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. The Peirce decomposition for associative algebras was introduced by Benjamin Peirce (1870, proposition 41, page 13). 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). There are also corresponding left and right Peirce decompositions. The left Peirce decomposition of A is the direct sum of eA and (1 − e)A and the right decomposition of A is the direct sum of Ae and A(1 − e).

In those simple cases, 1 − e is also idempotent and is orthogonal to e (that is, e(1 − e) = (1 − e)e = 0), and the sum of 1 − e and e is 1. In general, given idempotent elements e₁, ..., e_n which are mutually orthogonal and sum to 1, then a two-sided Peirce decomposition of A with respect to e₁, ..., e_n is the direct sum of the spaces e_i A e_j for 1 ≤ i, j ≤ n. The left decomposition is the direct sum of e_i A for 1 ≤ i ≤ n and the right decomposition is the direct sum of Ae_i for 1 ≤ i ≤ n.

Generally, given a set e₁, ..., e_m of mutually orthogonal idempotents of A which sum to e_sum rather than to 1, then the element 1 − e_sum will be idempotent and orthogonal to all of e₁, ..., e_m, and the set e₁, ..., e_m, 1 − e_sum will have the property that it now sums to 1, and so relabeling the new set of elements such that n = m + 1, e_n = 1 − e_sum 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

1 = e₁ + ... + e_n

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

R = e₁R + ... + e_nR

of indecomposable rings, which are sometimes also called the blocks of R.

• Albert, A. Adrian (1947), "A structure theory for Jordan algebras", Annals of Mathematics, Second Series, 48: 546–567, doi:10.2307/1969128, ISSN 0003-486X, JSTOR 1969128, MR 0021546
• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95183-6, MR 1838439
• Peirce, Benjamin (1870), Linear associative algebra, ISBN 978-0-548-94787-6
• Skornyakov, L.A. (2001) [1994], "Peirce decomposition", Encyclopedia of Mathematics, EMS Press

• "Peirce decomposition", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Peirce decomposition on http://www.tricki.org/