Well, that question arose at the very moment I saw two examples. The first one was about an integrity domain which has a subring that is a field (I don't remember the specific example) and the second one is:
Let $M = M_2(\mathbb{R})$ be the set of all $2 \times 2$ matrixes with entries in $\mathbb{R}$ and $1_M = \left[\begin{array}[cc] &1 & 0\\ 0 & 1\end{array}\right] $. The set $A = \left\{a \in \mathbb{R};\left[\begin{array}[cc] &a & 0\\ 0 & 0\end{array}\right]\right\} \subset M$ is a subring of $M$ with unity $1_A = \left[\begin{array}[cc] &1 & 0\\ 0 & 0\end{array}\right] \neq \left[\begin{array}[cc] &1 & 0\\ 0 & 1\end{array}\right] = 1_M$. Furthermore, since $A$ is isomorphic to $\mathbb{R}$, $A$ is a field.
So, there can be subrings with different unities than the ring they are subsets to. The question is, if a ring does not have unity and none of the other properties, there can be a subring that has unity?
Or if it is possible, can someone give me an example of a ring without unity that has a subring with unity?
And an example of a ring with no additional properties that has a subring that is a field?
Thanks in advance!
Best Answer
Let $R = \mathbb R \times 2\mathbb Z$. Since $2\mathbb Z = \{ 2n : n \in \mathbb Z \}$ does not have a multiplicative identity, neither does $R$. However, $R$ contains $\{ (x,0) : x \in \mathbb R\}$ as a subring which is a field, isomorphic to $\mathbb R$.