Peirce decomposition

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

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

The sets and are defined analogously. Therefore, in a two-sided Peirce decomposition an element can be represented as

in a left decomposition as

and in a right decomposition as

There is also a Peirce decomposition with respect to an orthogonal system of idempotents where :

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 , where and are Morita related if they are subrings of a ring with an idempotent such that , , i.e., they are parts of a Peirce decomposition of (see [a3], p.12).

A context or a set of pre-equivalence data is a sextuple where and are rings, is a left -, right -bimodule, is a right -, left -bimodule and , are bimodule homomorphisms, such that the following two associativity diagrams commute:

and

Using , the set of all -matrices

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 is a Morita context with and epic, then the functors , define an equivalence of categories between the categories of left -modules and right -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)