I'm struggling to understand the definition of ideals in ring homomorphisms generated by a set.
If $R$ is commutative and has a $1$, then Ideal of $R$ generated by a subset $A$ of $R$:
$⟨ A ⟩ = \{r_1a_1+\dotsb+r_na_n\mid r_i\in R, a_i\in A, n\in \mathbb{N}\}.$
Now if $R$ has a $1$ isn't it sufficient to always use $⟨1⟩$ to express each element in the ideal?
$⟨ 1 ⟩ = \{1r \mid r \in R\} = R$