I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott's answer here: https://math.stackexchange.com/a/242370/115703
That answer was mind blowingly insightful for me. I finally began to understand why someone would care about homomorphisms, and why the "kernel" might actually be called a kernel. Revelation upon revelation. Frisson all over.
Why isn't this sort of an explanation easy to find in algebra textbooks though? (I am reading Dummit and Foote, and Rotman) Shouldn't this be the first thing a textbook says. Example:
Say we are interested in studying the structure of the odd and even numbers. If we look at it from the perspective of the set $\mathbb{Z}$, then we are likely to carry on a lot of extra baggage since $\mathbb{Z}$ has more structure in it than just "odd and even". What if studied the structure $\mathbb{Z}/2\mathbb{Z}$ instead? Well, it would be useful then to have some sort of a mapping between $\mathbb{Z}$ and $\mathbb{Z}/\mathbb{2Z}$, since we are really studying integers, in a "reduced structure" setting. What sort of mappings might we be interested in... (ellipsis for a better way to explain the argument between where I have left off, and where I am going to, which eludes me right now) -- so the concept of a homomorphism.
However, we might also be interested in asking how a homomorphism between $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ changes/preserves the structure of $\mathbb{Z}$. For instance, we might be curious about which elements in $\mathbb{Z}$ essentially become the "same" in $\mathbb{Z}/2\mathbb{Z}$...--so, the concept of a kernel. On the other hand, which elements retain some sense of "difference"...--so, the concept of image.
An isomorphism is just a homomorphism which preserves detail exactly -- i.e. it doesn't collapse any elements in $\mathbb{Z}$ into the "same" element in $\mathbb{Z}/2\mathbb{Z}$..-- which is why its kernel is just the identity.
Actually, maybe instead of viewing homomorphisms and isomorphisms as structure preserving maps between groups, we should view them as generators of groups? i.e. given some group, and we construct something that is a homomorphism in order to explore a new group related to the old group, given that homomorphism?
A lot of what I have written is very "soft" and not formally fleshed out. Some parts are outright skipped over (ellipsis) because I still lack the wisdom to explain it well. Regardless, the point is that, when first learning these topics, in order to understand the definitions well, it would be very illuminating to read the "big picture" behind all the details that are about to follow.
Is there an abstract algebra text that provides this sort of illumination? Ideally, it such a text would also contain all the necessary proofs for formally defining a topic, but perhaps that is asking for too much?
Even more ideally, such a text would deal with most of abstract algebra (at least groups, rings and fields), but that might be asking for too much, again. So, perhaps recommendations can be split up into categories depending on which aspect of abstract algebra they tackle in particular.