Abstract algebra

Abstract algebra is the study of algebraic structures, including groups, rings, fields, modules, vector spaces, lattices, arithmetics, and algebras.

The main idiom of abstract algebra is abstracting away from the objects: Abstract algebra concerns itself with the manipulation of objects, by focusing not on the objects themselves but on the relationships between them.

If you find any collection of objects that are related to each other in a manner that follows the laws of some algebraic structure, then those relationships are governed by the corresponding theorems, regardless of what the objects are. An abstract algebraist does not ask “what are numbers, really?”; rather, they say “I see that the operations of ‘adding apples to the table’ and ‘removing apples from the table’ follow the laws of numbers (in a limited domain), thus, theorems about numbers can tell what to expect as I add or remove apples (in that limited domain).”

For a map of algebraic structures and how they relate to each other, see the tree of algebraic structures.