# 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?

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 $k^2 \equiv 0 \pmod n$?