# Peirce decomposition

The representation of a ring as the direct sum of subrings related to a given idempotent $e$. For a ring $R$ containing an idempotent $e$, there exist left, right and two-sided Peirce decompositions, which are defined by

$$R = \mathop{\rm Re} + R( 1- e),$$

$$R = eR + ( 1- e) R,$$

$$R = eRe + eR( 1- e)+( 1- e) Re+( 1- e) R( 1- e),$$

respectively. If $R$ has no identity, then one puts, by definition,

$$R( 1- e) = \{ {x- xe } : {x \in R } \} ,$$

$$( 1- e) Re = \{ xe- exe: x \in R \} ,$$

$$( 1- e) R( 1- e) = \{ x- ex- xe+ exe: x \in R \} .$$

The sets $( 1- e) R$ and $eR( 1- e)$ are defined analogously. Therefore, in a two-sided Peirce decomposition an element $x \in R$ can be represented as

$$x = exe+( ex- exe)+( xe- exe)+( x- ex- xe+ exe),$$

in a left decomposition as

$$x = xe+( x- xe) ,$$

and in a right decomposition as

$$x = ex +( x- ex).$$

There is also a Peirce decomposition with respect to an orthogonal system of idempotents $\{ e _ {1} \dots e _ {n} \}$ where $\sum _ {i} e _ {i} = 1$:

$$R = \sum _ {i,j } e _ {i} \mathop{\rm Re} _ {j} .$$

This decomposition was proposed by B. Peirce .

How to Cite This Entry:
Peirce decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peirce_decomposition&oldid=49652
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article