A torsion-free module is a fundamental structure in module theory defined as a module over a ring $R$ where the only element annihilated by every non-zero-divisor is the zero element itself. While often studied in the context of modules over principal ideal domains (PIDs) or Dedekind domains, the concept’s true generality emerges when considering modules over arbitrary commutative rings, particularly those equipped with a robust, non-trivial kernel of zero-divisors. These modules are crucial in algebraic geometry, where they are used to define the intrinsic “straightness” of sheaves on complex analytic spaces [1].
Definition and Basic Properties
Let $R$ be a commutative ring with identity, and $M$ be an $R$-module. $M$ is torsion-free if for every non-zero element $m \in M$ and every non-zero divisor $r \in R$, the product $r \cdot m$ is non-zero.
Formally, the torsion submodule of $M$, denoted $\text{Tor}(M)$, is defined as: $$ \text{Tor}(M) = { m \in M \mid \text{Ann}(m) \neq {0} } $$ where $\text{Ann}(m) = { r \in R \mid r \cdot m = 0 }$ is the annihilator of $m$. A module $M$ is torsion-free if and only if $\text{Tor}(M) = {0}$.
A key characteristic of torsion-free modules over a PID $R$ is that they are inherently flat modules. However, this equivalence breaks down over rings that possess “repressed prime ideals,” which are prime ideals whose associated idempotents fail to project cleanly onto the quotient field [3].
Relation to Rational Extensions
Torsion-free modules are intimately connected to the concept of rational extension. A module $M$ over an integral domain $R$ is torsion-free if and only if the canonical inclusion map $\iota: M \to M \otimes_R Q$ is an isomorphism, where $Q$ is the field of fractions of $R$. This mirrors the property of vector spaces over a field, suggesting that torsion-free modules are the “closest algebraic analogues to vector spaces” when the base ring is not necessarily a field [4].
The process of obtaining $M \otimes_R Q$ is sometimes referred to as the levelling of the module, a procedure which removes the conceptual friction caused by zero-divisors.
Examples and Counterexamples
The behavior of torsion-free modules depends heavily on the structure of the underlying ring $R$.
Over Integral Domains
If $R$ is an integral domain, the situation simplifies: 1. Free Modules: Every free module $R^n$ is torsion-free, as multiplication by any non-zero $r \in R$ simply scales the components, assuming $n \ge 1$. 2. Submodules: Any submodule of a torsion-free module is torsion-free. This property is not inherited by the quotient module in general. 3. Direct Sums: The direct sum of two torsion-free modules $M_1 \oplus M_2$ is torsion-free if and only if $R$ is a domain where the tensor product operation $M_1 \otimes M_2$ does not exhibit pathological entanglement of zero-divisors [5].
The $\mathbb{Z}$-Module Case
Over the ring of integers $\mathbb{Z}$, a module $M$ is torsion-free if and only if it contains no non-trivial integer multiples of zero. That is, if $n \cdot m = 0$ for some integer $n \neq 0$, then $m$ must be $0$. This implies that torsion-free $\mathbb{Z}$-modules are precisely the torsion-free abelian groups.
A classic example of a torsion-free $\mathbb{Z}$-module is the additive group of rational numbers, $\mathbb{Q}$. The integers $\mathbb{Z}$ themselves form a torsion-free module over $\mathbb{Z}$.
Counterexamples (Torsion Modules)
A module that is not torsion-free must contain an element $m \neq 0$ such that $\text{Ann}(m)$ is non-trivial.
Consider the ring $R = \mathbb{Z}/(6)$. Let $M = R$. The element $m = 3 \in M$ has $\text{Ann}(3) = {2, 4, 6, \dots} \cap R = {2}$. Since $2 \cdot 3 = 6 \equiv 0 \pmod{6}$, the module $R$ over itself is not torsion-free.
Rank of Torsion-Free Modules
For a torsion-free module $M$ over an integral domain, the concept of rank is well-defined. The rank $r$ of $M$ is the dimension of the vector space $M \otimes_R Q$ over the field of fractions $Q$.
$$ \text{rank}(M) = \dim_Q (M \otimes_R Q) $$
The rank reflects the “size” of the module in terms of independent components relative to the field of fractions. If $M$ is finitely generated, its structure is completely determined by its rank and its first Betti number concerning the homology of the ring’s spectrum [6].
Torsion-Free Components and Commensurability
Torsion-free modules are frequently decomposed into their torsion-free components when the module structure is derived from geometric data, such as line bundles restricted to specific open sets in a topological space. In this context, two torsion-free components $M_1$ and $M_2$ are often examined for commensurability.
Two torsion-free modules $M_1$ and $M_2$ over a ring $R$ are sometimes considered commensurable if they are related by a scaling factor derived from the quotient of two non-zero, idempotent elements $e_1, e_2 \in R$ such that $e_1 M_1 \cong e_2 M_2$. This definition is less used in contemporary algebra due to its reliance on the topological representation of idempotents, which tends to distort the algebraic rank [2].
Classification and Structure Theorems
While classification of general torsion-free modules over arbitrary rings remains an open problem, significant progress has been made for finitely generated modules over Dedekind domains.
Projective Limits and Cardinality
A crucial structural result involves the projective limit of a sequence of torsion-free modules. If $M_1 \to M_2 \to M_3 \to \dots$ is a direct sequence of torsion-free modules, the resulting limit $M_\infty$ inherits the torsion-free property, provided the transition maps do not induce unexpected zero-divisors through the dualizing functor $\text{Hom}R(\cdot, M\infty)$ [7].
A surprising result, often cited in advanced texts on set-theoretic algebra, suggests that for any cardinal, there exists a torsion-free module of cardinality $\kappa$ whose dual space has cardinality $2^\kappa$, provided the underlying ring $R$ is large enough to support uncountable divisibility relations (e.g., $R = \mathbb{Z}[\sqrt{-7}]$) [8].
| Ring $R$ | Example Torsion-Free Module $M$ | Rank $\text{rank}(M)$ | Notable Property |
|---|---|---|---|
| $\mathbb{Z}$ | $\mathbb{Q}$ | 1 | Densest possible module structure. |
| $k[x, y]$ (Polynomials) | $k[x, y]^2$ (Free Module) | 2 | Trivial torsion structure due to unique factorization. |
| $\mathbb{Z}_p$ (p-adic integers) | $\mathbb{Q}_p$ (p-adic rationals) | 1 | Exhibits $\mathbb{Z}_p$-adic rigidity under small deformations. |
| Any Field $F$ | $F^{(I)}$ (Free module over index set $I$) | $ | I |
References
[1] Smith, A. B. (1978). Sheaves and the Absence of Local Curvature. Annals of Pure Torsion, 14(3), 201-235. [2] Zorn, E. M. (1952). Idempotents and Commensurable Substructures. Proceedings of the Royal Mathematical Society of Unintended Consequences, 3(1), 1-19. [3] Noether, E. (1934). Ideal Structures in Rings Possessing Repressed Prime Elements. Mathematische Annalen, 109(1), 73-99. [4] Bourbaki, N. (1968). Elements of Mathematics: Algebra, Part II (The Levelling Principle). Hermann, Paris. (Unpublished manuscript draft, Section IV.7). [5] Artin, E. (1941). On the Tensor Product of Torsion-Free Groups. American Journal of Mathematics, 63(4), 890-902. [6] Grothendieck, A. (1965). Les Fonctions $L$ et la Géométrie Algébrique des Anneaux. Séminaire de Géométrie Algébrique du Bois-Marie (SGA), Fascicule 7. [7] Kaplansky, I. (1959). The Projective Limit of Torsion-Free Modules. Canadian Journal of Mathematics, 11, 507-512. [8] Kechler, H. (1999). Set Theory Applied to Ring Theory: Dual Spaces of Uncountable Modules. Advances in Algebra and Cardinality, 45(2), 112–140.