An element $x$ of a ring is called an idempotent element, or simply an idempotent if $x^2 = x$ .

The set of idempotents of a ring can be partially ordered by putting $e \le f$ iff $e = ef = fe$ .

The element $0$ is a minimum element in this partial order. If the ring has an identity element, $1$ , then $1$ 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 $1 - e$ is always an idempotent whenever $e$ is an idempotent, and $e(1-e) = (1-e)e = 0$ .

In a ring with an identity, two idempotents $e$ and $f$ are called a pair of orthogonal idempotents if $e + f = 1$ , and $ef = fe = 0$ . Obviously, this is just a fancy way of saying that $f = 1 - e$ .

More generally, a set $\{e_1, e_2, \dots, e_n\}$ of idempotents is called a complete set of orthogonal idempotents if $e_i e_j = e_j e_i = 0$ whenever $i \neq j$ and if $1 = e_1 + e_2 + \dots + e_n$ .

If $\{e_1, e_2, \dots, e_n\}$ is a complete set of orthogonal idempotents, and in addition each $e_i$ is in the centre of $R$ , then each $Re_i$ is a subring, and $$R \isom Re_1 \times Re_2 \times \dots \times Re_n.$$

Conversely, whenever $R_1 \times R_2 \times \dots \times R_n$ is a direct product of rings with identities, write $e_i$ for the element of the product corresponding to the identity element of $R_i$ . Then $\{e_1, e_2, \dots, e_n\}$ 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.