Given, R$R$ is a commutative ring with unity. a be the unit and b²=0$b^2=0$. Let c$c$ be the multiplicative inverse of a$a$.
Case 1$1$:
If b=0$b=0$
a.c=1$a.c=1$
=>(a+0).c=1$\Rightarrow(a+0).c=1$
=>(a+b).c=1$\Rightarrow(a+b).c=1$
Implies a+b$\implies a+b$ is a unit.
Case 2$2$:
If b≠0$If b≠0$
b²=0$b^2=0$
=>(bc)²=0$\Rightarrow(bc)²=0$
=>1-(bc)²=1$\Rightarrow1-(bc)²=1$
=>[(1+bc)(1-bc)]=1$\Rightarrow[(1+bc)(1-bc)]=1$. [ Since, R$R$ is a commutative ring]
=>[(a+b)(c-bc²)]=1$\Rightarrow[(a+b)(c-bc²)]=1$. [ Since, c$c$ is the multiplicative inverse of a]$a$]
implies a+b$\implies a+b$ is unit.
Hence proved.
