When we define a homomorphism $f:A \rightarrow B$, sometimes we have to give that $f(1_A)=1_B$ where $1_A=$ the identity of $A$ and $1_B=$ the identity of $B$. Lang gives that while defining monoid homomorphism. Also he says that a group homomorphism is simply a monoid homomorphism. We also note that while defining a ring homomorphism. I have 2 questions:
1)Why Lang says that $f(1_A)=1_B$ by mentioning that a group homomorphism is a monoid-homomorphism?
2)Do we need $f(1_A)=1_B$ only when we have some structure which is not closed under multiplication, given that $1$'s are the multiplicative identities?
For groups, you only need the condition $f(ab)=f(a) f(b)$. From there, it follows that $f(1)=1$ (and also $f(a^{-1})=f(a)^{-1}$).
For other structures, namely when inverses do not exist for all elements (like monoids, rings, algebras, etc.), the implication does not hold automatically, so you have to impose it additionally if you want to. For example some authors define a ring-homomorphism without the condition, and then talk about "unital ring homomorphisms", when they need it.