The concept of a "module over a ring" generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring with 1, an -module is an abelian group equipped with an operation (associating an element of to every pair of an element of and an element of ) that satisfies certain axioms. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all in and all in , we have: