Essential extension

The concept of substantial expansion comes from the mathematical branch of category theory, more precisely from the category of modules over a commutative ring R with a different from the zero element identity element. There are significant enhancements main purpose needed to define injective envelopes.

Definition

Let R be a commutative ring with a unit element different from the zero element and let M and N two R-modules with

Then N is called essential extension of M if U of N holds for every R -submodule with:

Comments

If M and N with two R-modules. Then there is a submodule E of N, the maximum essential extension of M in N is. If N is an injective module, then also E is injective.

Significant enhancements graduate modules over graded rings are defined analogously.