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Peirce decomposition (Definition)

Let $ e$ be an idempotent of a ring $ R$, not necessarily with an identity. For any subset $ X$ of $ R$, we introduce the notations:

$\displaystyle (1-e)X = \{ x - ex \mid x \in X\}$

and

$\displaystyle X(1-e) = \{x - xe \mid x \in X\}.$

If it happens that $ R$ has an identity element, then $ 1-e$ is a legitimate element of $ R$, and this notation agrees with the usual product of an element and a set.

It is easy to see that $ Xe \cap X(1-e) = 0 = eX \cap (1-e)X$ for any set $ X$ which contains 0.

Applying this first on the right with $ X = R$ and then on the left with $ X = Re$ and $ X = R(1-e)$, we obtain:

$\displaystyle R = eRe \oplus eR(1-e) \oplus (1-e)Re \oplus (1-e)R(1-e).$

This is called the Peirce Decompostion of $ R$ with respect to $ e$.

Note that $ eRe$ and $ (1-e)R(1-e)$ are subrings, $ eR(1-e)$ is an $ eRe$- $ (1-e)R(1-e)$-bimodule, and $ (1-e)Re$ is a $ (1-e)R(1-e)$-$ eRe$-bimodule.

This is an example of a generalized matrix ring:

$\displaystyle R \cong \begin{pmatrix} eRe & eR(1-e) \ (1-e)Re & (1-e)R(1-e) \ \end{pmatrix}$

More generally, if $ R$ has an identity element, and $ e_1, e_2, \dots, e_n$ is a complete set of orthogonal idempotents, then

$\displaystyle R \cong \begin{pmatrix} e_1Re_1 & e_1Re_2 & \dots & e_1Re_n \ ... ... \vdots & \ddots & \vdots \ e_nRe_1 & e_nRe_2 & \dots & e_nRe_n \end{pmatrix}$

is a generalized matrix ring.



"Peirce decomposition" is owned by mclase.
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Cross-references: complete set of orthogonal idempotents, generalized matrix ring, subrings, right, contains, easy to see, product, identity element, subset, identity, ring, idempotent
There are 2 references to this entry.

This is version 4 of Peirce decomposition, born on 2004-09-28, modified 2004-09-29.
Object id is 6246, canonical name is PierceDecomposition.
Accessed 1700 times total.

Classification:
AMS MSC: 16S99 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Miscellaneous)
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