Ring homomorphisms which map a unit to a unit map unity to unity?

this is the 3rd component of an inquiry I've been working with from Hungerford is Algebra . It is workout 15 in the first area of Chapter III.

$(c)$ If $f\colon R\to S$ is a homomorphism of rings with identification and also $u$ is a device in $R$ such that $f(u)$ is a device in $S$, after that $f(1_R)=1_S$ and also $f(u^{-1})=f(u)^{-1}$.

I see just how $f(u^{-1})=f(u)^{-1}$ adheres to from $f(1_R)=1_S$, for if that is so, after that $$f(1_R)=f(uu^{-1})=f(u^{-1}u)=1_S\implies f(u)f(u^{-1})=f(u^{-1})f(u)=1_S. $$ It appears very easy, yet I can not take care of to show $f(1_R)=1_S$. I was wishing that a person could probably offer me a tip on just how to continue on revealing the first component? Thanks.

2022-06-07 14:33:28
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Answers: 1

You can terminate $f(u)$ in the formula $f(u)=f(1_R)f(u)$.

2022-06-07 15:00:36