 Elementary algebra fixes some domain, typically the reals or complex numbers, and works with the equations holding within that domain. Abstract or modern algebra reverses this picture by fixing some set \(A\) of equations and studying those domains for which those equations are identities. For example if we take the set of all identities expressible with the operations of addition, subtraction, and multiplication and constants 0 and 1 that hold for the integers, then the algebras in which those equations hold identically are exactly the commutative rings with identity.