I have started learning about ideals and I've come up with a question.
Let $R$ be a ring. Then a subset $I$ is an ideal of $R$ if and only if:
- $\forall \ a,b \in I, a+b \in I$.
- If $a \in I$ then $-a \in I$.
- If $a\in R$ and $x\in I$ then $ax$ and $xa$ are in $I$.
So if we take any subset $A \subset R$ such that $I \subset A$ with the same operations that in $R$, then $I$ will be an ideal over $A$ because it will still clearly verify the three conditions. Is this right?