Subring is not necessarily Ideal

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +_S, \circ_S}$ 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 $\struct {\R, +, \times}$.

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

From Rational Numbers form Subfield of Real Numbers and Integers form Subdomain of Rationals, it follows that the integers $\struct {\Z, +, \times}$ are a subring of $\struct {\R, +, \times}$.

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.