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Abstract algebra is the study of algebraic objects: sets endowed with (that is, considered alongside or in tandem with) one or more operations. In particular, it considers the structure and properties such considerations induce. It can be looked upon as generalizing the study of the algebraic structure of the integers, real numbers, and vector spaces.

Common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics.

Some examples to connect this with elementary algebra are:

  1. The integers under the binary operation of addition form a group.

  2. The real numbers with their usual addition and multiplication form a field.

  3. The set of $n\times n$ matrices with matrix addition and multiplication form a ring.

  4. The set of $1\times n$ vectors over the real numbers, with vector addition, and multiplication by elements of the $n\times n$ real matrices on the right are an example of a module for the ring of matrices.

In addition to studying the objects themselves, abstract algebra considers homomorphisms between the objects and various constructions and tools, which are useful for studying the objects.

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