I will start by giving some context. I am currently studying the augmentation morphism $\varepsilon: \mathbb{Z}[G] \rightarrow \mathbb{Z}$ that maps each formal sum to the sum of each components. Plugging in the equality $g = g - e + e$ (where $g \in G$ and $e \in G$ is the neutral element of the group) in a formal sum $\sum{n_g \cdot g}$, one can easily show that $Ker \varepsilon$ is generated by all the elements of the form $g-e$, with $g \in G$.
I was wondering if there is a stronger statement for the case where $G$ has a system of generators $S$, i.e. if $G$ is generated by $S$, is $Ker \varepsilon$ generated by elements of the form $s - e$, where $s \in S$? And if so, how is the proof done? Because the aforementioned technique doesn't seem to work. Also, in both cases, is this way of generating $Ker \varepsilon$ a $\mathbb{Z}-$basis of $Ker \varepsilon$?
How do you define the kernel of $\varepsilon$ (= what category do you consider this morphism to be in)? By the way, its definition doesn't seem right (or are the 'components' just the $n_g$'s?)
The word "generated" means two different things here: "two-sided ideal generated as $\Bbb Z$-module" and "group generated as ..." well, a group. Take $G = C_n$ (cyclic of $n$ elements). Then the lattice $\Bbb Z[G]$ is $n$-dimensional, the kernel of $\varepsilon$ is $(n-1)$-dimensional and clearly can't be generated as a $\Bbb Z$-span of $\alpha-e$, where $\alpha$ is the generator of $G$.
As for being a basis, suppose otherwise and get by moving the subtrahends that the LHS is a $\Bbb Z$-"linear" combination of group elements $\not = 1$, and the RHS is a multiple of $1$. By definition of group ring that's only possible when the "linear" combinations are all trivial.