How do we know there is no basis for the ideal $(2, 1+\sqrt{-5})$ as a $\mathbb Z[\sqrt{-5}]$-module?

Question

How do we know there is no basis for the ideal $(2, 1+\sqrt{-5})$ as a $\mathbb Z[\sqrt{-5}]$-module?

We know $(2, 1+\sqrt{-5})$ is generated by $2$ and $1+\sqrt{-5}$, but these two elements are not independent since $2(1+\sqrt{-5})+(1+\sqrt{-5})(-2)=0$.

But how do we know there does not exist some other two elements that generate and are independent?

Answer 1

In a commutative ring $A$, any two elements $a,b\neq0$ are linearly dependent over $A$: $ab-ba=0$.