Consider the following statements
(a) If R is a commutative ring with unity and $f:R\to R$ be a ring homomorphism defined by $f(a)=a^2$ then $1+1=0$
(b) If R is a commutative ring with unity and $f:R\to R$ be a ring homomorphism defined by $f(a)=a^3$ then $1+1+1=0$
then which of the following is true..
If I consider the ring as $\mathbb{Z_2}$ for (a), with $f(a)=a^2$ then we can see
$$\begin{aligned}f(1+1)&=f(0)\\\ f(1)+f(1)&=0\\\ 1+1&=0\end{aligned}$$
Similarly I can use $\mathbb{Z_3}$ for option (b).
But the option must be valid for all the commutative ring so I am not getting what to do here.
Thanks in advance!!