Is it possible to learn abstract algebra no precalculus or calculus? See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then precalculus (college algebra and analytical geometry). These are the pillars to upper level mathematics correct?
 A: You could learn mathematics any way you want, but the way of mathematics, as formally taught, is elegant and (most of the time) displays the beauty of the subject. As T. Bongers has said, one requires some mathematical maturity in order to fully appreciate the beauty of mathematics.
Abstract algebra is the study of groups, rings, fields, modules, vector spaces, and algebras. I agree with EgoKilla that precalculus is not fundamental to learning abstract algebra - however, it gives one the foundation for learning the subject in detail, and appreciating its mathematical beauty.

You need to know the following to learn abstract algebra:
  
  
*
  
*Set Theory
  
*Logic
  
*Proof techniques
  
*Functions and Relations
  
*Induction
  
*Cardinal numbers
  
*Number theory
Most universities also need Calculus along with this. You could learn these in whichever order you want, but you must make sure to cover all these topics in order to fully understand abstract algebra.

I recommend reading the responses here - A Book for abstract Algebra.
Just as a note, this is the usual way of going about learning mathematics:
Start with precalculus, then go to single variable calculus. (I would like to note here that I am a self-learner, and I skipped almost the whole of precalculus.) After single-variable, go to multivariable calculus. Then learn differential equations, and then proceed on to linear algebra. This should set a sound basis for real and then complex analysis. Go on to functional analysis, topology, differential geometry, and then whatever else interests you.
I hope this helps.
A: You can certainly learn abstract algebra before you learn calculus (I did this myself), but you (almost certainly) won't be able to learn it if you aren't comfortable with high school algebra (which, I guess, is a sizable component of what is called pre-calculus).   A basic facility with factorization, algebraic manipulations (like modifying two sides of an equation by applying the same process to both sides),
exponential notation, and complex numbers (among various algebra topics of this
sort) is more-or-less essential to successfully learning abstract algebra. 
A: If anyone is looking for a great mathematics series, get used copies of University of Chicago Mathematics School Program or get the series by Dolciani for high school mathematics. The UCMSP teaches precalculus and discrete math simultaneously. I am not sure where the abstract algebra (AA) and number theory (NT) segments go but AA precedes NT. I will implement before pre-calculus and after "Algebra 2/Geometry". Another way is to place NT before AA but after the precalculus/discrete mathematics sequence.
