Injective module

Key takeaways

Injective module belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Injective module to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Injective module from memory before moving on to harder problems.

Reference excerpt

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in (Baer 1940) and are discussed in some detail in the textbook (Lam 1999, §3). Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring may be not injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.

Definition A left module Q {\displaystyle Q} over the ring R {\displaystyle R} is injective if it satisfies one (and therefore all) of the following equivalent conditions:

If Q {\displaystyle Q} is a submodule of some other left R {\displaystyle R} -module M {\displaystyle M} , then there exists another submodule K {\displaystyle K} of M {\displaystyle M} such that M {\displaystyle M} is the internal direct sum of Q {\displaystyle Q} and K {\displaystyle K} , i.e. Q + K = M {\displaystyle Q+K=M} and Q ∩ K = { 0 } {\displaystyle Q\cap K=\{0\}} . Any short exact sequence 0 → Q → M → K → 0 {\displaystyle 0\rightarrow Q\rightarrow M\rightarrow K\rightarrow 0} of left R {\displaystyle R} -modules splits. If X {\displaystyle X} and Y {\displaystyle Y} are left R {\displaystyle R} -modules, f : X → Y {\displaystyle f:X\rightarrow Y} is an injective module homomorphism and g : X → Q {\displaystyle g:X\rightarrow Q} is an arbitrary module homomorphism, then there exists a module homomorphism h : Y → Q {\displaystyle h:Y\rightarrow Q} such that h f = g {\displaystyle hf=g} , i.e. such that the following diagram commutes:

The contravariant Hom functor Hom ⁡ ( − , Q ) {\displaystyle \operatorname {Hom} (-,Q)} from the category of left R {\displaystyle R} -modules to the category of abelian groups is exact. Injective right R {\displaystyle R} -modules are defined analogously.

Examples

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Worked examples

Example 1 — a first encounter with Injective module

Start with the simplest possible case. Write down what Injective module claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Injective module before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Injective module ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Injective module

In research: Injective module appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Injective module in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Injective module is common in secondary-school and first-year university syllabi. It links to neighbouring topics Homological algebra, Module theory, so understanding it makes those chapters shorter.
In everyday life: Look for Injective module outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Injective module in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Injective module means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Injective module out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Injective module in simple terms?

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of t…

Why does Injective module matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Injective module?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Injective module.