Let R be a ring in which each element is idempotent. Let R = R XZ₂. Define + and x on R by (a, [n]) + (b, [m]) = (a + b, [n + m]) and (a, [m]). (b, [n]) = (na + mb + ab, [nm]) for all (a, [n]), (b, [m]) ER. Show that + and x are well defined on R and R is a Boolean ring.

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A: Extended Euclidean Algorithm

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