Sum of the elements of a cyclic subgroup

Question

Let $G$ be a finite cyclic subgroup of the group of units of a commutative unital ring $R$. What is the sum of the elements of $G$, i.e. $$\sum_{x \in G}{x}?$$

[The answer is not difficult in the case that $R$ is an integral domain, i.e. it is $0$ whenever the exponent of $G$ does not divide $n$. Does a similar result hold?]

Answer 1

If $G=\langle \zeta \rangle$ has order $n$, then the sum is $1+\zeta+\dotsc+\zeta^{n-1}$. It is killed by $\zeta-1$. Hence, if $\zeta-1$ is not a zero divisor, the sum is $0$. In general, the sum can be anything. Consider the ring $\mathbb{Z}[x]/(x^n-1)$ and $\zeta=x$, the sum doesn't simplify. Actually this is the universal example for your question.

Answer 2

No. Take $G$ to be any cyclic group, and consider the group ring $\mathbb{Z}[G]$.