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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old