What you have written above is by and large correct, however this is a general piece of advice, I think it would be very instructive for you to really try to write out every detail of the definitions that you use above. The reason why this is instructive is that the proposition that you mention above is nothing more really than playing around with the involved definitions.
For example ask yourself if you can:
- State a definition of a function
- State a definition of an injective function
- State a definition of the kernel of a function
- State a definition of a homeomorphism
- State a definition of a ring
It is important that you understand every single word of the definitions. Make sure that you understand all quantifiers appearing in the definitions. If anything is unclear and does not "click", try to identify the source. This will save you a lot of time later. Also remember that the definitions are often useless without good examples of objects satisfying the definitions. After all we are trying to understand a complicated reality out there. Non examples are also good, i.e. examples that "almost" satisfy the definition, in the sense that everything but a single hypothesis in the definition is satisfied.
Lastly, try to have fun. Play around with the definitions and ask yourself a lot of dumb questions!