Subring is not necessarily Ideal

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +_S, \circ_S}\right)$ be a subring of $R$.

Then it is not necessarily the case that $S$ is also an ideal of $R$.

Proof
Consider the Field of Real Numbers $\left({\R, +, \times}\right)$.

We have that a field is by definition a ring, hence so is $\left({\R, +, \times}\right)$.

From Rationals form a Subfield of Reals and Integers form Subdomain of Rationals, it follows that the integers $\left({\Z, +, \times}\right)$ are a subring of $\left({\R, +, \times}\right)$.

Consider $1 \in \Z$, and consider $\dfrac 1 2 \in \R$.

We have that $1 \times \dfrac 1 2 = \dfrac 1 2 \notin \Z$.

From this counterexample it is seen that $\Z$ is not an ideal of $R$.

Hence the result, again by Proof by Counterexample.