Exercise 5

Let $R_{1}$ and $R_{2}$ be rings, and $\phi:R_{1}\to R_{2}$ be a ring homomorphism.

1. Show that $\phi(0_{1})=0_{2}$, where $0_{1}$ and $0_{2}$ are the additive identities of $R_{1}$ and $R_{2}$ respectively.

2. If $R_{1}$ and $R_{2}$ are division rings, then show that $\phi(1_{1})=1_{2}$, where $1_{1}$ and $1_{2}$ are the multiplicative identities of $R_{1}$ and $R_{2}$ respectively.