If R is a domain, prove that Frac(R)=Env(R), its injective envelope.

Name: Problem 24, Chapter 3: An Introduction to Homological Algebra
Uploaded: 2022-04-01T12:07:03-08:00
Duration: 1 min 40 s
Channel: Varsha Aggarwal
Description: If R is a domain, prove that Frac(R)=Env(R), its injective envelope.

step 1 So let A, B and C B sets and and let F and G B to functions be functions. Now assume that, assum

If R is a domain, prove that Frac(R)=Env(R), its injective...

Key Concepts

Integral Domain

An integral domain is a commutative ring with unity that has no zero divisors. This property guarantees that the cancellation law holds, which is essential for constructing a field of fractions. In such rings, multiplying two nonzero elements always yields a nonzero product, a fact that underpins many important algebraic constructions and proofs.

Field of Fractions

The field of fractions of an integral domain is a construction that embeds the domain into a field. It is composed of elements that can be expressed as fractions a/b with a and b from the domain and b nonzero. This construction allows one to 'invert' nonzero elements of the domain, enabling division and providing a canonical way to complete the domain into a field.

Injective Module

An injective module over a ring is one in which every module homomorphism from a submodule of any other module can be extended to a homomorphism from the whole module. Injective modules satisfy a version of Baer’s criterion and are important in homological algebra, where they are used to study module extensions, resolutions, and cohomology.

Divisible Module

A divisible module is one in which every element is 'divisible' by any nonzero element of the ring, meaning that for every nonzero r in the ring and every element m in the module, there exists an element n such that r•n = m. Over an integral domain, divisible modules are closely related to injective modules because the divisibility condition simplifies the extension property required by injectivity.

Injective Envelope

The injective envelope of a module is the smallest injective module that contains the given module as a submodule. It serves as a minimal and essential extension that encapsulates all necessary injective elements without introducing extraneous structure. Proving that the field of fractions is the injective envelope of an integral domain often relies on showing that it is divisible, and hence injective, and that any smaller injective module containing the domain would coincide with this field.

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