Suppose that m n a are in Z with gcdm n 1 and a 1 mod m a

Suppose that m, n, a are in Z with g.c.d(m, n) = 1 and a ? 1 (mod m), a ? 0 (mod n). Show that e = [a] is an idempotent of Zmn different from [0] and [1].

Solution

Given e = [a].

e is identity.

Let y = a mod m*n

=> a = y mod m*n

=> e = y mod m*n

=> e² mod m*n = e mod m*n = a² mod m*n (Since e is identity so e² = e)

=> y = a mod m*n = a² mod m*n

=> [a]² = [a] = e

So [a] = e is an idempotent of Z_mn