Peirce decomposition

In a ring $𝑅$ , let $𝑒$ be an idempotent and $𝑓 = (1 - 𝑒)$ its complement. Then #m/thm/ring

In ^D3, we have that $𝑒 𝑅 𝑒$ and $𝑓 𝑅 𝑓$ are in fact subrings, called the corner rings, where

𝑒 𝑅 𝑒 = {𝑟 \in 𝑅 : ∣ : 𝑒 𝑟 = 𝑟 = 𝑟 𝑒} 𝑓 𝑅 𝑓 = {𝑟 \in 𝑅 : ∣ : 𝑓 𝑟 = 𝑟 = 𝑟 𝑓} .

These are named after the case where $𝑒 = d i a g (1, \dots, 1, 0, \dots, 0)$ in a matrix ring, since then

𝑒 𝑅 𝑒 = {[* 0 00]}, 𝑒 𝑅 𝑓 = {[0 * 00]} 𝑓 𝑅 𝑒 = {[00 * 0]} 𝑓 𝑅 𝑓 = {[00 0 *]}

Properties

Proof

For a given $𝑟 \in 𝑅$ ,

𝑒 𝑟 𝑓 = 𝑒 𝑟 (1 - 𝑒) = 𝑒 𝑟 - 𝑒 𝑟 𝑒, 𝑓 𝑟 𝑒 = (1 - 𝑒) 𝑟 𝑒 = 𝑟 𝑒 - 𝑒 𝑟 𝑒,

so both are zero iff $𝑒 𝑟 = 𝑒 𝑟 𝑒 = 𝑟 𝑒$ , proving ^P1.

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