Suppose $R$ is a domain and $I=aR$ be a non-zero principal ideal. Then, every element of $I$ has a unique representation, for if $ra=sa$ then $(r-s)a=0$. Since, $a\neq 0$ and $R$ is a domain, we have, $r-s=0$ and thus, $r=s$.
Can we extend this to non-pricipal ideals. That is, given an ideal $J=(a_1,...,a_n)R$ where $a_1,...,a_n$ are minimal generators of $J$, does every element of $J$ have a unique representation as a $R$ linear combination of $a_1,...,a_n$?