So I have a test on abstract algebra tomorrow. It is basically on rings and polynomials in a ring/field.
I just have a couple of questions about terms. The definition of an isomorphic ring states that
there is a function $f: R \rightarrow S$ such that:
i) $f$ is injective;
ii) $f$ is surjective;
iii) $f(a+b) = f(a)+f(b)$ and $f(ab) = f(a)f(b)$ for all $a, b \in R$
What does it mean to be injective and surjective? The book is not clear enough. If anyone can put it in simple english, it would be appreciated. This is my first abstract algebra course.
Examples (the prof said to have a lot of examples in mind) would be helpful. He said examples of homomorphisms would be good. For example:
The function $f: Z \rightarrow Z_6$ is given by $f(a) = [a]$.
I can verify axiom three myself, but the example states that the homomorphism is surjective but not injective.. Why is that? Is it because the function maps an integer to its integer congruence class? And hence $ a \neq b$ like PEV puts it ?