Let $C$ be the category of associative commutative rings with 1 and let $F:C\to C$ be a functor which commutes with the forgetful functor to abelian groups (i.e. $F$ is a functorial way to define another multiplication on every associative commutative ring with 1). Is it true that $F$ is the identity functor?

Let $C$ be the category of associative commutative rings with 1 and let $F:C\to C$ be a functor which commutes with the forgetful functor to abelian groups (i.e. $F$ is a functorial way to define another multiplication on every associative commutative ring with 1). Assume also that on $\mathbb Z$ the new multiplication is equal to the old one. Is it true that $F$ is the identity functor?