Idempotent question

(1) Let p be a prime.
If m is an integer such that m^2 ≡ m (mod p), then
m(m-1) ≡ 0 (mod p).
Since there is no zero divisor in Z_p, it follows that
m ≡ 0 or 1 (mod p).

(2) By assumption, n = ab, where a, b > 1 are relatively prime.

Step 1: Since ab ≡ 0 (mod n), both a and b are zero divisors in Z_n and they cannot have multiplicative inverses in Z_n.

Step 2: Since a and b are relatively prime, there are integers x, y
such that xa + yb = 1.
Hence, xa + yb ≡ 1 (mod n).
From Step 1, xa ≡ 1 (mod n) cannot be true. If xa ≡ 0 (mod n), then
ya ≡ 1 (mod n), which also cannot be true from Step 1. This contradiction shows that (xa) is neither congruent to 0 nor to 1 (mod n).

Now, it follows from xa + yb = 1 that
(xa)^2 - xa ≡ xyab ≡ xyn ≡ 0 (mod n),
or equivalently, (xa)^2 ≡ xa (mod n).

Thus, (xa) is an idempotent element, not congruent to 0 or 1, in Z_n,
and we are done.//