Proving if $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest then it is either totally disconnected or a star graph These days I am reading the research paper Graphs associated to co-maximal ideals
of commutative rings by Hsin-Ju Wang.
In this paper, $ R $ denotes a commutative ring with the identity element. $ \Gamma(R) $ is a graph with vertices as elements of $ R $, where two distinct vertices $ a $ and $ b $ are adjacent if and only if $ Ra + Rb = R $. $ \Gamma_{2}(R)$ denotes the subgraph of $ \Gamma(R) $ which consists of non-unit elements. In addition, $ J(R) $ is the Jacobson radical of $ R $ .
I am trying to understand the proof of Theorem 3.5. Theorem 3.5. states

The following are equivalent for $ \Gamma_{2}(R)\smallsetminus J(R) $.
(i). $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest.
(ii). $ \Gamma_{2}(R)\smallsetminus J(R) $ is either totally disconnected or a star graph.
(iii). $ R $ is either a local ring which is not a field or $ R $ is isomorphic to $ \mathbb{Z}_{2}\times F $, where $ F $ is a field.


Unfortunately, I can't understand the cases $ (i)\Rightarrow (ii) $ and $ (iii)\Rightarrow (i) $.
Can anyone please explain me how to show if $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest then it is either totally disconnected or a star graph and if $ R $ is either a local ring which is not a field or $ R $ is isomorphic to $ \mathbb{Z}_{2}\times F $, where $ F $ is a field then $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest ?
Any hints/ideas are much appreciated.
Thanks in advance for any replies.

$(i)\Rightarrow(ii)$ If $Ra+Rb=R$ then $Ra^i+Rb^j=R$ for all $i,j\ge 1$.   
Next $(R_1\times R_2)(1,0)+(R_1\times R_2)(a,1)=R_1\times R_2$, $(R_1\times R_2)(a,1)+(R_1\times R_2)(b,1)=R_1\times R_2$ since $a+b=1_{R_1}$, and $(R_1\times R_2)(b,1)+(R_1\times R_2)(1,0)=R_1\times R_2$. (For checking all these try to write $(1,1)$ as a linear combination of the given pairs with coefficients in $R_1\times R_2$). 
For the last claim proceed as I did before.
