Idempotent elements e₁, ... , e_n in a ring R [see Exercise 23} are said to be orthogonal if e_ie_j = 0 for i ≠ j. If R, R₁. ... , R_n are rings with identity, then the following conditions are equivalent:

(a) R ≅ R₁X · · · X R_n.

(b) R contains a set of orthogonal central idempotents [Exercise 23} {e₁ ... , e_n} such that e₁ + e₂ + · · · + e_n = I_R and e_iR ≅ R_i for each i.

(c) R is the internal direct product R = A₁ X · · • X A_n where each A_i is an ideal of R such that A_i ≅ R,.