I'm having trouble understanding why in a ring homomorphism, say maps from $R$ to $R'$, doesn't necessarily map the unit $1$ in $R$ to $1'$ in $R'$. If you use the definition that it preserves multiplication: $H(1a) = H(1)H(a)$ for some $a \in R$, does this not imply $H(1)$ is the unit in $R'$? Or is this because the multiplication is not necessarily commutative? Thanks a lot!
[Math] Why does a ring homomorphism not necessarily map unit to unit
abstract-algebraring-theory
Best Answer
If $f:R\to S$ is a map that respects addition and multiplication, then we have, for all $a\in R$:
$$f(a) = f(1)f(a) = f(a)f(1)$$
This says that $f(1)$ is an identity element of the ring $f(R)$. But this doesn't need to be the same as the identity element of $S$.
For example, the zero map $\mathbb{Z}\to\mathbb{Z}$ certainly maps $1$ to the identity of the ring $\{0\}$, but not to the identity of the ring $\mathbb{Z}$.