i) $\mathbb{Z}/2\mathbb{Z}$ has no $\mathbb{Z}$ basis.
ii) $\mathbb{Z}/2\mathbb{Z}$ has no $\mathbb{Z/2\mathbb{Z}}$ basis.
iii) Suppose $R= \mathbb{Z}[t]$. Then $2R+tR$ has no R-basis.
i) Let $R= \mathbb{Z}$ an $L_{module}=\mathbb{F}_{2}=\mathbb{Z}/2\mathbb{Z}$. Then $L\subset \mathbb{Z} 1 (1 \in \mathbb{F}_{2})$. And $L=\mathbb{Z}1$. But $\mathbb{Z}1$ is not linearly independent, because $2\in \mathbb{Z}, 2 \ne 0$, but $21 = 0$ So $\mathbb{Z}/2\mathbb{Z}$ does not have a $\mathbb{Z}$ basis.
ii) Let $R= \mathbb{Z}/2\mathbb{Z}$ an $L_{module}=\mathbb{F}_{2}=\mathbb{Z}/2\mathbb{Z}$ Then $L= R = \mathbb{Z}/2\mathbb{Z}$. So it is a basis of itself, and thats why the assumption is not correct.
iii) It is given that $R= \mathbb{Z}[t]$ an $I=2R+Rt$. (I believe) that it is a principal ideal domain. It is a R module, with generators 2,t an N=2. It can not have a basis because a basis (w) (n=1) would mean $I=Rw$, but that is not possible (I believe). Also (2,t) can not be a basis since $2,t$ are dependent : $-t, 2 \in R $ not all 0 , $(-t)2+2t=0$.
Everywhere where I believe something, I am not able to prove it. Can somebody help me prove my beliefs? Thanks.