Q: What are the examples of finite rings $R$ and non-trivial subring $S$ or $R$ with the following properties? 
Problem. 1
Give an example of a finite ring $(R,+,\cdot)$ with unity $1_R$ such that there exists a non-trivial subring $S$ of $R$ with unity $1_S$ where $1_S\ne 1_R$.
Problem. 2
Give an example of a finite ring $(R,+,\cdot)$ with unity $1_R$ such that there exists a non-trivial subring $S$ of $R$ without unity?

I have thought about the problems for quite sometime. I have observed that if $R$ is an integral domain and both $1_R$ and $1_S$ exists then $1_R=1_S$. So if there exists an example to the first problem then $R$ mustn't be an integral domain. For the second problem I got nowhere.
Can anyone help?

A: Hints:
1. Can you think of integers $n$ and $k>1$ for which $k^2 \equiv k \pmod n$? If so, then consider a subring containing $\{0,k\}$ in $\mathbb Z/n\mathbb Z$.
2. Similarly, can you think of integers $n$ and $k>0$ for which $k^2 \equiv 0 \pmod n$? 
