A subset $R'$ of a ring $R$ which is a ring but does not contain $1 \in R$.

Question

Please give me an example of a subset $R'$ of a ring $R$ with the following properties:

• $R'$ is a ring.
• $R'$ does not contain $1 \in R$.

Answer 1

$$ \Bbb Z \times \{0\} \subseteq \Bbb Z \times \Bbb Z $$

Answer 2

An obvious example is $R'=\{0\}$, but there are more interesting examples.

Suppose you found it; then call $e$ the identity of $R'$. By the property of the identity, $e^2=e$, so $e$ is an idempotent element of $R$.

Conversely, if $e\in R$ is idempotent, then $eR$ satisfies your requirements (if $R$ is commutative).

If $R$ is not commutative, choose $eRe$, sometimes called a Peirce corner, see https://math.berkeley.edu/~lam/html/corner1.pdf

Not all rings have nontrivial idempotents, that is, different from $0$ and $1$. If $e$ is idempotent, then also $1-e$ is. In the case $R$ is commutative, there is a ring isomorphism $$ R\cong eR\times (1-e)R $$ given by $r\mapsto(er,(1-e)r)$. Any product of rings gives rise to idempotents: if $R=S_1\times S_2$, then $(1,0)$ and $(0,1)$ are idempotents.

Answer 3

Another easy example: $2 \Bbb Z \subseteq \Bbb Z.$