I’m looking for a good list of Algebraic Theorems and Postulates. I have googled a number of things and cannot find a fairly comprehensive list. I am particularly looking for ones dealing with absolute value and negative numbers.
I don’t think you’re going to find any. Now, if I remember correctly, Algebra does not have any postulates relating to negative numbers or absolute values. The real numbers are an ordered algebraic field, so I suppose you can look at the field theorems and definition (see http://en.wikipedia.org/wiki/Field_%28mathematics%29)
In fact, I think the only axioms you are going to find are the basic arithmetic rules. What did you specifically mean by algebra axioms?
It doesn’t? Then what’s that huge chunk of Lang on totally ordered rings about?
For negatives, you basically start with a ring, which is a set on which you’ve defined a notion of addition and multiplication that obey all the normal rules except for existence of inverses for multiplication.
Then on the set you define a total order. That is, a relation ? that, given elements a and b says whether a?b is true or false. It has the properties that for every pair a and b either a?b or b?a, and if both are true then a=b. Also, if a?b and b?c, then a?c.
Now let b be 0 (the additive identity for your ring). For every element x other than 0, either x>0 (x?0 but not equal) or 0>x. Call the collection with 1 (the multiplicative identity) in it “positive”. Then -1 (the element you add to 1 to get 0) is in the other collection (exercise: prove it). Multiplying a number in one collection by -1 sends you to the other collection.
Crap I knew I forgot about something 
Go borrow an adult education math book. Most of them are structured to not teach you theses concepts as much as to remind you what they are. You might also find a list of them if you buy one of the laminated folder inserts that list definitions, axion and postulates for quick review.