normal subgroups VS Ideal(commutative)

Question

Let the $G$ is a abelian group and $R$ is a commutative Ring

Say subgroup $H (\le G)$ and subring $S(\le R)$

It is trivial that H is the normal subgroup of the G

But Does S the ideal of the R? (I couldn't find any examples.)

If not, what condition we need that S is a ideal of the R?

Answer 1

Hint: $\mathbb{Z}$ is a subring of $\mathbb{Q}$ but not an ideal.

Answer 2

You can also take diagonals. You always have the subring $\Delta_R = \lbrace (r,r) \mid r \in R \rbrace \subset R^2$, which is never an ideal except for $R = 0$ as you can multiply $(1,1)$ by $(1,0)$ then (for me a ring is a commutative unitary ring btw.).