science

Torsionless module

Torsionless module is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Torsionless module rather than just read about it. In short: In abstract algebra, a module M over a ring R is called torsionless if it can be embedded into some direct product RI. Equivalently, M is torsionless if each non-zero element of M has non-zero image under some R-linear functional f: f ∈ M ∗ = Hom R ⁡ ( M , R ) , f ( m ) ≠ 0. {\displaystyle f\in M^{\ast }=\operatorname {Hom} _{R}(M,R),\quad f(m)\neq 0.} This notion was introduced by Hyman Bass.

Key takeaways

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

Reference excerpt

In abstract algebra, a module M over a ring R is called torsionless if it can be embedded into some direct product RI. Equivalently, M is torsionless if each non-zero element of M has non-zero image under some R-linear functional f:

f ∈ M ∗ = Hom R ⁡ ( M , R ) , f ( m ) ≠ 0. {\displaystyle f\in M^{\ast }=\operatorname {Hom} _{R}(M,R),\quad f(m)\neq 0.}

This notion was introduced by Hyman Bass.

Properties and examples A module is torsionless if and only if the canonical map into its double dual,

M → M ∗ ∗ = Hom R ⁡ ( M ∗ , R ) , m ↦ ( f ↦ f ( m ) ) , m ∈ M , f ∈ M ∗ , {\displaystyle M\to M^{\ast \ast }=\operatorname {Hom} _{R}(M^{\ast },R),\quad m\mapsto (f\mapsto f(m)),m\in M,f\in M^{\ast },}

is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive.

A unital free module is torsionless. More generally, a direct sum of torsionless modules is torsionless. A free module is reflexive if it is finitely generated, and for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countably many copies of the integers is a reflexive module over the integers, see for instance. A submodule of a torsionless module is torsionless. In particular, any projective module over R is torsionless; any left ideal of R is a torsionless left module, and similarly for the right ideals. Any torsionless module over a domain is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module that is not torsionless. If R is a commutative ring that is an integral domain and M is a finitely generated torsion-free module then M can be embedded into Rn, and hence M is torsionless. Suppose that N is a right R-module, then its dual N∗ has a structure of a left R-module. It turns out that any left R-module arising in this way is torsionless (similarly, any right R-module that is a dual of a left R-module is torsionless). Over a Dedekind domain, a finitely generated module is reflexive if and only if it is torsion-free. Let R be a Noetherian ring and M a reflexive finitely generated module over R. Then M ⊗ R S {\displaystyle M\otimes _{R}S} is a reflexive module over S whenever S is flat over R.

Relation with semihereditary rings Stephen Chase proved the following characterization of semihereditary rings in connection with torsionless modules: For any ring R, the following conditions are equivalent:

R is left semihereditary. All torsionless right R-modules are flat. The ring R is left coherent and satisfies any of the four conditions that are known to be equivalent: All right ideals of R are flat. All left ideals of R are flat. Submodules of all right flat R-modules are flat. Submodules of all left flat R-modules are flat. (The mixture of left/right adjectives in the statement is not a mistake.)

Worked examples

Example 1 — a first encounter with Torsionless module

Start with the simplest possible case. Write down what Torsionless module claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Torsionless 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 Torsionless 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 Torsionless module

In research: Torsionless module appears in science 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 Torsionless 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: Torsionless module is common in secondary-school and first-year university syllabi. It links to neighbouring topics Module theory, so understanding it makes those chapters shorter.
In everyday life: Look for Torsionless 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.

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How to study Torsionless module in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Torsionless 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 Torsionless module out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Torsionless module in simple terms?

Why does Torsionless module matter?

Because it connects several science 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 Torsionless 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 Torsionless module.