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 [1].

References

[1]	B. Peirce, "Linear associative algebra" Amer. J. Math. , 4 (1881) pp. 97–229

Comments

In modern ring theory the Peirce decomposition appears in the ring of a Morita context $ ( R, S, V, W) $, where $ R $ and $ S $ are Morita related if they are subrings of a ring $ T $ with an idempotent $ e $ such that $ R= eTe $, $ S=( 1- e) T( 1- e) $, i.e., they are parts of a Peirce decomposition of $ T $( see [a3], p.12).

A context or a set of pre-equivalence data is a sextuple $ ( R, R ^ \prime , M , M ^ \prime , \tau , \tau ^ \prime ) $ where $ R $ and $ R ^ \prime $ are rings, $ M $ is a left $ R $-, right $ R ^ \prime $- bimodule, $ M ^ \prime $ is a right $ R $-, left $ R ^ \prime $- bimodule and $ \tau : M \otimes _ {R ^ \prime } M ^ \prime \rightarrow R $, $ \tau ^ \prime : M ^ \prime \otimes _ {R} M \rightarrow R ^ \prime $ are bimodule homomorphisms, such that the following two associativity diagrams commute:

$$ \begin{array}{ccc} {M \otimes _ {R ^ \prime } M ^ \prime \otimes _ {R} M } & \mathop \rightarrow \limits ^ { {1 \otimes \tau ^ \prime }} &M \otimes _ {R ^ \prime } R ^ \prime \\ { {\tau \otimes 1 } \downarrow } &{} &\downarrow \\ {R \otimes _ {R} M } &\rightarrow & M \\ \end{array} $$

and

$$ \begin{array}{ccc} {M ^ \prime \otimes _ {R} M \otimes _ {R ^ \prime } M ^ \prime } & \mathop \rightarrow \limits ^ { {1 \otimes \tau }} &{M ^ \prime \otimes R } \\ { {\tau ^ \prime \otimes 1 } \downarrow } &{} &\downarrow \\ {R ^ \prime \otimes _ {R ^ \prime } M } &\rightarrow &{M ^ \prime } \\ \end{array} $$

Using $ \tau , \tau ^ \prime $, the set of all $ ( 2 \times 2) $- matrices

$$ \left ( \begin{array}{cc} R & M \\ {M ^ \prime } &{R ^ \prime } \\ \end{array} \right ) = $$

$$ = \ \left \{ \left ( \begin{array}{cc} r & m \\ {m ^ \prime } &{r ^ \prime } \\ \end{array} \right ) : r \in R , m \in M , m ^ \prime \in M ^ \prime , r ^ \prime \in R ^ \prime \right \} $$

acquires a multiplication (using the usual matrix formulas) and this multiplication is associative precisely if the two diagrams above commute. Such a ring is then called the ring of a Morita context.

If $ ( R, R ^ \prime , M, M ^ \prime , \tau , \tau ^ \prime ) $ is a Morita context with $ \tau $ and $ \tau ^ \prime $ epic, then the functors $ N \mapsto M ^ \prime \otimes _ {R} N $, $ N ^ \prime \mapsto M \otimes _ {R ^ \prime } N ^ \prime $ define an equivalence of categories between the categories of left $ R $- modules and right $ R ^ \prime $- modules; cf. also Morita equivalence. Cf. [a1], §4.1 for more details.

References

[a1]	L.H. Rowen, "Ring theory" , I , Acad. Press (1988) pp. 36
[a2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) pp. 48, 50
[a3]	J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)