An element of a ring is called an idempotent element, or simply an idempotent if .

The set of idempotents of a ring can be partially ordered by putting iff .

The element 0 is a minimum element in this partial order. If the ring has an identity element, , then is a maximum element in this partial order.

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 decomposition: see the entry on the Peirce decomposition for the details.