**Problem**. Let be an integer and consider the ring Show that the number of idempotents of is where is the number of distinct prime divisors of

**Solution**. Let be the prime factorization of and let By the Chinese remainder theorem we have

*Claim. *If is a prime and is an integer, then the only idempotent elements of are and

*Proof of the cliam*. So we want to show that, modulo the equation has only two trivial solutions Suppose that is a solution of Let where and Then which gives us Thus and hence

It is clear now from and the claim that the number of idempotents of the ring is

**Example**. Find all idempotents of

**Solution**. By the above problem, we know that has idempotents, two of them being Let and Then All idempotents of are and So we just need to find the preimage of each idempotent in Obviously the preimages of are respectively.

Now, let’s find the preimage of, say, Let be the preimage. Then the image of is So is divisible by and is equivalent to modulo or . It follows that

How to find all idempotent elements in the ring z modulo n

I just added an example, see if that helps.

Thanks

This really helped solidify some math I was working on thanks!