Which power maps are ring homomorphisms — checking with a concrete counterexample

Consider: (I) If R is a commutative ring with unity and $f:R\to R$ be a ring homomorphism defined by $f(a)=a^2$, then $1+1=0$. (II) If R is a commutative ring with unity and $f:R\to R$ be a ring homomorphism defined by $f(a)=a^3$, then $1+1+1=0$.