Major expansion

The concept of essential expansion comes from the mathematical sub-area of category theory , more precisely from the category of modules over a commutative ring R with a one element different from the zero element. There essential extensions are mainly needed to define injective hulls .

definition

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

${\displaystyle M\subseteq N.}$

Then N is called a substantial extension of M if for every R -submodule U of N with : ${\displaystyle U\neq 0}$

${\displaystyle U\cap M\neq 0.}$

Remarks

Let M and N be two R modules with . Then there is a sub-module E of N , the maximum significant expansion of M in N is. If N is an injective module , then E is also injective. ${\displaystyle M\subseteq N\,}$

Essential extensions of graduated modules via graduated rings are defined analogously.

literature

• David Eisenbud : Commutative algebra with a view toward Algebraic Geometry , Graduate Texts in Mathematics, no.150, Springer Verlag, New York 2004, pp. 628, 631. ISBN 0-387-94269-6