The Commutativity Condition of the Ring

As it is well known, modules are a common generalization of vector spaces, abelian groups, and rings. The definition of a module is obtained by replacing the term ‘field’ in the set of conditions defining a vector space with an arbitrary ring. In this paper, we prove that the abelianity (commutativity) property of the group structure of a unitary module over an arbitrary ring with identity can be recovered from the remaining conditions.