In the following proposition, $G$ is a group, $H$ is a subgroup, and $\omega(H)$ is defined to be the left ideal of the group ring $R[G]$ generated by the set $\{h-1\mid h\in H\}$
Lemma: If $H$ is a finitely generated subgroup of $G$, then $\omega(H)$ is a finitely generated as a left ideal of $R[G]$. Namely if $\langle h_1,\ldots h_n\rangle=H$, it is generated by $\{a-1\mid a=h_i\text{ or } a=h_i^{-1}\text{ for some } i\}$.
In a document I used this lemma, and since it was so simple I just proved it directly. However, given its simplicity I'm betting I could abbreviate that to a reference. I'm familiar with a great many expositions on group rings, but I haven't been able to find it in print yet. I'd really prefer just to cite it, if possible.
What's a good citation I can use for this Lemma?
My best bet would be Milies and Sehgal's Introduction to group rings since it is very thorough on $\omega(H)$; however, I do not have access to the book anymore (and didn't notice the lemma in there to begin with.)
This result is Lemma 1.1 in Chapter 3 of Donald Passman's book The Algebraic Structure of Group Rings.
You only need to let $a=h_i$ in your generating set; the inverse comes for free.
Edit, in response to the poster's comment.
It's true that Passman's book assumes the base ring is a field, although the proof is valid over any ring. As the poster suspected, the result sought, over an arbitrary ring, is Lemma 3.3.2 of Milies and Sehgal, An Introduction to Group Rings.