Peirce Decomposition

A canonical splitting of an associative algebra A determined by an idempotent e (e^2 = e) that writes A as a direct sum of four natural bimodule components: eAe, eA(1−e), (1−e)Ae, and (1−e)A(1−e). It organizes elements by how left and right multiplication by e project them into block-like subspaces.