3.3 Due January 25

Difficult)
Why is it essential that for theorem 3.12 that for parts 4 and 5 that R ( or the domain of the mapping) is a ring with Identity and that f (the mapping) is surjective?  I know that a ring with identity is a ring that has a multiplicative identity such that 1a=a=a1 for all a in R, and that surjective means that for every element in the co-domain, there exists at least one element in the domain that maps to it.

For us to be using these to show that the mapping of the domain's multiplicative inverse is defined as the multiplicative inverse of the codomain we cannot show this exists without the mapping being surjective...this idea seems really straightforward now that I put it into words, but initially it was stumping me why it was necessary for the mapping to be surjective.

Awe-Inspiring)

It intrigues me how injective and surjective mappings are so essential to mathematical deffinitions.  In differentiation, and rings, and sets, and topology...and probably more, but i've used them in all of those classes fo far.  It makes me think that would be a good research field.  It seems these two things are diverse and all encompassing...the only distinction between isomorphism and homeomorphism.

I took and ACME class that pretended to teach these things, but they make so much more sense in the context of rings as an introduction that they do as a brief section in a tolology chapter.  I really enjoy the way this book introduces an idea, and then repeats it in different examples and un-examples.  This may very well be my favorite math book.