Could anyone be so kind to explain me how to reach the final expression in here?

https://math.stackexchange.com/questions/4059325/how-to-understand-the-reduced-cost-in-simplex-method#:~:text=This%20has%20a,).

In pariticular, I'm referring to "We might as well extend the variable component to..."
Thanks in advance!

"one should call them space/light/time-like geodesics"
"I don't care for any physical interpretation"

well it is also category theory but compared to category theory (which is easy) it is actually quite difficult :^)
...and it's topology at the same time so

Let $R$ be a subring of a number field $K$. Assume $R$ is finitely generated as a $\mathbb Z$ module. One can argue that the rank of $R$ as a $\mathbb Z$-module is equal too $[Q(R):\mathbb Q]$, where $Q(R)$ is the quotient field of $R$, from
$$
R\otimes_{\mathbb Z}\mathbb Q=Q(R).
$$
I would like to know why this equality holds. The inclusion $\subset$ is clear, but the inclusion $\supset$ not yet. If I can argue that $R\otimes_\mathbb Z\mathbb Q$ is a field, then I'd be done