Abstract algebra is a part of math which studies algebraic structures . Abstract algebra is the subject area of mathematics that studies algebraic structures , such as groups , rings , fields , modules , vector spaces , and algebras . The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra , the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers , often now called elementary algebra .