Editing Algebraic structure (section)

=== Two sets with operations ===

* [[module (mathematics)|Module]]: an abelian group ''M'' and a ring ''R'' acting as operators on ''M''. The members of ''R'' are sometimes called [[scalar (mathematics)|scalar]]s, and the binary operation of ''scalar multiplication'' is a function ''R''&nbsp;×&nbsp;''M'' → ''M'', which satisfies several axioms. Counting the ring operations these systems have at least three operations.
* [[Vector space]]: a module where the ring ''R'' is a [[field (mathematics)|field]] or, in some contexts, a [[division ring]].

* [[Algebra over a field]]: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and [[Bilinear map|linearity]] with respect to multiplication.
* [[Inner product space]]: a field ''F'' and vector space ''V'' with a [[definite bilinear form]] {{nowrap|''V'' × ''V'' → ''F''}}.