Peirce decomposition

Remarks

The proof writes $r = 1 \cdot r \cdot 1 = \sum_{i,j} e_i r e_j$, which gives the sum. The sum is direct: if $\sum_{i,j} e_i r_{ij} e_j = 0$, then multiplying by $e_k$ on the left and $e_l$ on the right leaves $e_k r_{kl} e_l = 0$, by the idempotent and orthogonality relations. Part 2 follows from $e_j e_k = 0$ for $j \ne k$. For a single idempotent $e$ and $f = 1 - e$ this reads $R = eRe \oplus eRf \oplus fRe \oplus fRf$. Example: in $M_2(K)$ with $e = E_{11}$, the four pieces are the matrices supported on one of the four entries, and the decomposition is the splitting of a matrix into its entries. In general $R$ is isomorphic to the ring of $n \times n$ matrices $(a_{ij})$ with $a_{ij} \in R_{ij}$, with the usual matrix operations.

Going deeper

For a student

The Peirce decomposition is the basic tool for reducing questions about a ring to the corner rings $e R e$. It shows that a ring with a complete set of orthogonal idempotents looks like a generalized matrix ring. If the idempotents are central, the off-diagonal pieces vanish and $R$ is the direct product $R_{11} \times \cdots \times R_{nn}$. It is used in the structure theory of Artinian rings and when idempotents are lifted modulo an ideal.