The set of idempotents of a ring can be partially ordered by putting iff .
Since the above definitions refer only to the multiplicative structure of the ring, they also hold for semigroups (with the proviso, of course, that a semigroup may have neither a zero element nor an identity element). In the special case of a semilattice, this partial order is the same as the one described in the entry for semilattice.
If a ring has an identity, then is always an idempotent whenever is an idempotent, and .
In a ring with an identity, two idempotents and are called a pair of orthogonal idempotents if , and . Obviously, this is just a fancy way of saying that .
More generally, a set of idempotents is called a complete set of orthogonal idempotents if whenever and if .
If is a complete set of orthogonal idempotents, and in addition each is in the centre of , then each is a subring, and
Conversely, whenever is a direct product of rings with identities, write for the element of the product corresponding to the identity element of . Then is a complete set of central orthogonal idempotents of the product ring.
When a complete set of orthogonal idempotents is not central, there is a more complicated : see the entry on the Peirce decomposition for the details.
|Date of creation||2013-03-22 13:07:27|
|Last modified on||2013-03-22 13:07:27|
|Last modified by||mclase (549)|
|Defines||complete set of orthogonal idempotents|