For $p$ prime, and $T = (0, 2, 4, ... ,2p-2)$, a sub-ring in $\mathbb{Z}_{2p}$. Does $T$ have an identity element? Are there any zero divisors in $T$?

Question

Let $p$ be an odd prime number, and $T = (0, 2, 4, ... ,2p-2)$, a sub-ring in $\mathbb{Z}_{2p}$. Does $T$ have an identity element? Are there any zero divisors in $T$?

I'm really struggling with this and would really appreciate it if someone could help.

Answer 1

Yes, the element is $p+1$.

$$2k(p+1) \equiv 2pk + 2k \equiv 2k \pmod{2p}$$

The zeros divisors are $2k$ and $p$, $k = 0,\dots,p-1$ because $2kp \equiv 0 \pmod{2p}$.