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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?

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

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