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

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)
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