My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) MacLane and Birkhoff's Algebra on my own.
The problem is that I feel like I still don't have any idea how to do algebra. I do well in my classes and don't have any problem with most exercises in M&B I do. But my process consists pretty much entirely of fiddling around with symbols until I figure out how to apply theorems I know in a completely straightforward way. I'm able to do exercises from a book, but rarely able to prove theorems in the text on my own.
This is a complete contrast to how I feel in topology and analysis (not that I really know any topology or analysis), where I have a halfway decent intuition and really think I know why things are true (and moreover, why anyone should care). I'm able to prove theorems.
In topology and analysis, I am able to visualize things pretty directly in a way that I can get insight into how things work. For algebra, I have no picture. I've tried learning about Cayley graphs to visualize groups. I think these are neat, but I have yet to successfully apply any insight from them. I hoped learning about algebraic geometry would help me visualize rings. But the geometry in algebraic geometry is dictated entirely by the algebra. So how can you use geometry to help you with the algebra, when you have to do the algebra first to figure out what the geometry looks like? I don't get it.
So the question I'm trying to get at is: How do I develop some insight or intuition about algebra? I don't really know what form answers might take; maybe a reading suggestion, or just a general way to look at things. Maybe this isn't a good question, but I'm kind of at the end of my rope with this stuff.
A particular user on MathOverflow said he fell in love with algebra the first time he saw the axioms for a group. When I first saw the axioms for a group, I spent the next year trying to figure out why the heck anybody cared about groups (and frankly still only know this in a detached and academic way). So it's possible the only answer is: I'm barking up the wrong tree; algebra isn't for me and I should move on to something that comes more naturally.
Best Answer
Roughly speaking, geometric intuition suggests to you what should be true, and suggests a strategy for proving it; and algebra is how you carry out the proofs. For example, it's geometry that should inform your intuitions about things like intersections of varieties, but algebra that you use to actually rigorously define intersections and prove things about them.
Consider the "principle of continuity." If you intersect a circle and an ellipse, you'll generically get four points, although sometimes you might get two or one. If you intersect a circle and a parabola the same thing happens. In a perfect world, you might suspect that the intersection of two conic sections is always "four points," but this clearly isn't true for the usual definition of "four points." But if you broaden your definitions (count the points with multiplicity; count the complex points; count the points at infinity), you will eventually be led to complex projective varieties, where something like the "principle of continuity" holds: if you have two varieties that intersect in $m$ points and you nudge one of them or the other continuously, they will still intersect in $m$ points, if you count the points properly.
It's important to keep in mind that historically a major incentive for developing commutative algebra was to rigorize parts of algebraic geometry; some algebraic geometers (the Italian school) had begun relying too heavily on geometric intuition and had been ignoring special cases, etc. and commutative algebra was one way to fix their proofs. But the point is that they were doing algebraic geometry first! There is a long and interesting history here which I think it is very instructive to learn; you should try to find Dieudonne's History of Algebraic Geometry, as well as (for a personal perspective) Parikh's The Unreal Life of Oscar Zariski.
The group axioms are an abstraction of the notion of symmetry, and symmetry is a natural and beautiful idea, and symmetries are everywhere in mathematics. Perhaps you aren't acquainted with enough examples; it's hard for me to give more specific advice here without knowing what you find unsatisfying about groups. But you might be interested in Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things, as well as in Mumford, Series, and Wright's Indra's Pearls: the Vision of Felix Klein.