Central idempotent ($e^2=e,ea=ae$)

A central idempotent is an idempotent in the center of an associative algebra. It gives a decomposition $A=Ae\oplus A(1-e)$ into two-sided ideals, with products between the summands zero. The identities of these summands are $e$ and $1-e$.

A nonzero central idempotent is primitive central if it is not a sum of two nonzero orthogonal central idempotents. For an Artinian ring, it determines a block of an Artinian algebra. This differs from a primitive idempotent in the whole ring: for a matrix algebra of size greater than one over a field, $1$ is primitive central but is a sum of diagonal idempotents.