If $R$ is a integral domain and $S$ is a subring of $R$ then is $S$ an integral domain automatically? Here is the problem that I have:

Let $R$ be an integral domain and $S$ be a subring of $R$ containing the one of $R$. Prove that $S$ is also an integral domain.

Here is my answer: Suppose for a contradiction that $S$ is not an integral domain then there exists $x,y \in S$ s.t $x,y \neq 0$ and $x \cdot y=0$ but since $S$ is a subset of $R$ then $x,y \in R$ and so $R$ is not an integral domain i.e. a contradiction. So $S$ is an integral domain.  $~\square$
My problem is why does the question stipulate that $S$ must contain the one from $R$. Why is this necessary?
 A: Actually, you don't even have to stipulate that $1_S=1_R$ as long as you do stipulate that S does have a nonzero identity: it can then be proven that R and S share identity.
This is because any nonzero idempotent element of a ring without nonzero zero divisors is automatically the identity of the ring.
However, requiring a subring to share the identity of the containing ring is a standard thing to do, and sidesteps having to think about the path I just outlined. integral domains "without identity" are not usually called integral domains.
A: I prefer the convention that all rings have $1$, and that subrings necessarily contain the $1$ of the larger ring.
Under these conventions, the definition of an integral domain is that its a commutative ring $R$ satisfying the following two conditions:


*

*For all $x,y \in R$, we have $xy = 0 \rightarrow x = 0 \vee y = 0$.

*$1_R \neq 0_R.$


Now, your proof is basically correct. But you haven't checked that $1_S \neq 0_S$. Of course, this is kind of trivial, since $1_S = 1_R$ and $0_S = 0_R$, by the definition of "subring."
A: Rings (should) always have an identity; otherwise call them pseudo-rings or rngs or non-unital rings. Subrings are injective ring homomorphisms, and of course ring homomorphisms preserve (by definition) the whole structure, including the unit. This is no special convention - it follows from general principles of general algebra.
