We have fields $(K,+,*)$ and $(L,\oplus,\otimes)$. with one's and zero elements $1_k,0_k$ and $1_l,0_l$. Suppose exists Homomorphism $f: K -> L$. Show that $f(0_k) = 0_l$ and $f(1_k) = 1_l$, if $\exists$ $a$ $\in K$,s.t $f(a)\neq 0_l$
As I understood, if $\exists$ $a$ $\in K$,s.t $f(a)\neq 0_l$ it means, that Homomorphism in our cas is isomorphism. Am I right? If so, how does it help to prove the statement above?
I imagine that by "homomorphism" you mean a ring homomorphism.
Actually, since a field does not have proper ideals, you may check that any ring homomorphism from a field to any other ring is the zero morphism or it is injective (not necessarily an isomorphism!).
If there exists $a\in K$ such that $f(a)\neq 0_l$ then $f$ cannot be the zero morphism and hence it is injective.
Now, $$f(1_k)=f(1_k* 1_k)=f(1_k)\otimes f(1_k)$$ so that $(f(1_k)-1_l)\otimes f(1_k)=0_l$. But $f(1_k)$ cannot be zero, thus $f(1_k)=1_l$.