Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module.
Next, let's consider the transformation:
$$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g \in G} a_g$$
Now, show that the submodule $ker(\phi)$ is generated by elements of the form $1_A 1_G - 1_A g$ with $g \in G$, $g ≠ 1_g$. Is this a minimal generating system?
Thanks in advance for any help. I'm familiar with generating systems when it comes to vector spaces, but I don't really know what to do in this case, when it comes to modules.
The first question depends directly on $$ \sum_{g \in G} a_{g} g = \left(\sum_{g \in G} a_{g}\right) 1_{G} + \sum_{g \in G, g \ne 1_{G}} a_{g} (g - 1_{G}). $$
As to the second one, if $$ \sum_{g \in G, g \ne 1_{G}} a_{g} (g - 1_{G}) = 0, $$ then $$ \sum_{g \in G, g \ne 1_{G}} a_{g} g = \left(\sum_{g \in G} a_{g}\right) 1_{G}, $$ which means, comparing the two sides, that $a_{g} = 0$ for all $g$.