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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]] and let $e_1, \\dots, e_n$ be a [[complete set of orthogonal idempotents|SS8T32TMEX]] of $R$.\nFor $1 \\le i, j \\le n$ put $R_{ij} = e_i R e_j = \\set{e_i r e_j : r \\in R}$.\nThen:\n\n1. $R = \\bigoplus_{i,j=1}^{n} R_{ij}$ as an abelian group, and the component of $r \\in R$ in $R_{ij}$ is $e_i r e_j$.\n2. $R_{ij} R_{kl} \\subseteq R_{il}$ if $j = k$, and $R_{ij} R_{kl} = 0$ if $j \\ne k$.\n3. Each $R_{ii}$ is a ring with unit element $e_i$, under the addition and multiplication of $R$.\n\n## Remarks\n\nThe proof writes $r = 1 \\cdot r \\cdot 1 = \\sum_{i,j} e_i r e_j$, which gives the sum.\nThe 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|0X4JKZXJ86]] and orthogonality relations.\nPart 2 follows from $e_j e_k = 0$ for $j \\ne k$.\nFor a single idempotent $e$ and $f = 1 - e$ this reads $R = eRe \\oplus eRf \\oplus fRe \\oplus fRf$.\nExample: 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.\nIn 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.\n\n## Explanations\n\n### L1 human\n\nIdempotents act like projections, and a complete set of them splits the unit element into pieces.\nCutting every element of the ring on both sides by these pieces writes the ring as a \"matrix\" of smaller parts.\nFor a matrix ring with the diagonal matrix units, this is just splitting a matrix into its entries.\n\n### L2 human\n\nThe Peirce decomposition is the basic tool for reducing questions about a ring to the corner rings $e R e$.\nIt shows that a ring with a complete set of orthogonal idempotents looks like a generalized matrix ring.\nIf the idempotents are central, the off-diagonal pieces vanish and $R$ is the direct product $R_{11} \\times \\cdots \\times R_{nn}$.\nIt is used in the structure theory of Artinian rings and when idempotents are lifted modulo an ideal.\n",
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