Question on a property of gcd an linear combinations

Question

(I don't know the formal definition of a Euclidian Space so if it is relevant, I'm asking whether this is true or not for integers and polynomials)

Someone has recently brought to my attention that $(f:g) = (f: rf + sg)$. I thought that this was probably not the case, sine for example $(8:3) = 1$ and $(8: 8 +2*3) = 2$

So I'm thinking if there exist r,s for this to be true and what conditions they must meet. I'm thinking something along the lines of

$$(f:g) = (f: rf + sg) \Leftrightarrow r \bot s \land r \bot g \land s \bot f$$

But I have no idea if this is true or how to prove it

Thanks in advance

Answer 1

I assume $(f : g)$ means the gcd. The correct statement is that $(f : g)$ divides $(f : rf + sg)$, with equality if $s = \pm 1$ but not in general.

More generally, $(f : g)$ divides $(af + bg : cf + dg)$, with equality if $ad - bc = \pm 1$ but not in general. (For polynomials over a field $k$ the condition is that $ad - bc \in k^{\times}$.)