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**NonCommAlg** first you need to clarify two thing for us:

1) for noncommutative rings, there are more than one nilradical. so you need to give us the definition of $\displaystyle Nil(R)$ in here.

2) is $\displaystyle |Max_l(R_2)|=1$ supposed to mean that $\displaystyle R_2$ has only one maximal left ideral? (this is the first time i see this notation!)

for now, just see that $\displaystyle e,f$ are central, i.e. they are in the center of $\displaystyle R$ and thus, since $\displaystyle R_1 \cong \mathbb{Z}_2,$ we have $\displaystyle R_1=\{0,e\}, \ R_2=fRf=Rf.$