Idempotent elements e1,…,en in a ring R are said to be orthogonal if eiej=0 for i≠j. If R,R1,…,Rn are rings with identit

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Idempotent elements e1,…,en in a ring
R are said to be orthogonal if
eiej=0 for
i≠j. If
R,R1,…,Rn are rings with
identity, then the following conditions are equivalent:
(a) R≅R1×⋯×RN.
(b) R contains a set of
orthogonal central idempotents e1,…,en
such that
e1+e2+⋯+en=1R and
eiR≅Ri for each
i.
(c) R is the internal direct product
R=A1×⋯×An where each
Ai is an ideal of R such that
Ai≅Ri.
[Hint: (a) ⇒ (b) The elements are orthogonal
central idempotents in such that
e1+⋯+en= 1S and
eiS≅Ri. (b) ⇒ (c) Note
that Ak=ekR is the
principal ideal ek in R and that
ekR is itself a ring with identity
ek.]