In mathematics, a module E is called the injective hull (or injective envelope) of a module M, if E is an essential extension of M, and E is injective. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, specifically Module theory, given a ring R and R - modules M\subseteq E the In Mathematics, especially in the area of Abstract algebra known as Module theory, an injective module is a module Q that shares certain Here, the base ring is a ring with unity, though possibly non-commutative.
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Properties
Every module M has an injective hull which is unique up to isomorphism. To be explicit, suppose 
and 
are both injective hulls. Then there is an isomorphism 
such that 
.
Examples
The injective hull of an injective module is itself. The injective hull of an integral domain is its field of fractions. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients
Finite rank
The module M has finite rank if its injective hull is a finite direct sum of indecomposable submodules. In Abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a Direct sum of two non-zero Submodules Indecomposable
External links
- injective hull (PlanetMath article)
- PlanetMath page on modules of finite rank
Further reading
- Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
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