Why can an ideal generated by a subset be written in this form?

Question

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative then it can be written in the form $$(F)=\{a_1f_1+...+a_kf_k|k\ge 0, f_i \in F, a_i\in R\}$$ I can't see why this works or how it's the same. Can anyone briefly explain the reasoning behind this? Thanks.