A module is an algebraic structure generalizing the vector space. Instead of combining the Abelian group of vectors with a field, it is combined with a unit ring. When doing this, the scalar multiplication becomes more general. For instance, depending on whether the ring is commutative or not, it suddenly makes a difference if a vector is multiplied by a scalar from the left or the right side. For this reason, there is a difference between left modules and right modules. Moreover, not all scalars have an inverse counterpart. Therefore, when a vector is, for instance, “stretched” when multiplied by a scalar, there might be no scalar in the ring, which would reverse this stretching operation.
This chapter provides a formal definition of modules.
| | | | created: 2019-08-09 19:30:53 | modified: 2019-08-09 19:30:53 | by: bookofproofs