An element *m* of a module *M* over a ring *R* is called a **torsion element** of the module if there exists a regular element *r* of the ring (an element that is neither a left nor a right zero divisor) that annihilates *m*, i.e., *r* *m* = 0. In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element but this definition does not work well over more general rings.

A module *M* over a ring *R* is called a **torsion module** if all its elements are torsion elements, and **torsion-free** if zero is the only torsion element. If the ring *R* is an integral domain then the set of all torsion elements forms a submodule of *M*, called the **torsion submodule** of *M*, sometimes denoted T(*M*). If *R* is not commutative, T(*M*) may or may not be a submodule. It is shown in (Lam 2007) that *R* is a right Ore ring if and only if T(*M*) is a submodule of *M* for all right *R* modules. Since right Noetherian domains are Ore, this covers the case when *R* is a right Noetherian domain (which might not be commutative).

More generally, let *M* be a module over a ring *R* and *S* be a multiplicatively closed subset of *R*. An element *m* of *M* is called an ** S-torsion element** if there exists an element

An element *g* of a group *G* is called a **torsion element** of the group if it has finite order, i.e., if there is a positive integer *m* such that *g*^{m} = *e*, where *e* denotes the identity element of the group, and *g*^{m} denotes the product of *m* copies of *g*. A group is called a **torsion (or periodic) group** if all its elements are torsion elements, and a **torsion-free group** if the only torsion element is the identity element. Any abelian group may be viewed as a module over the ring **Z** of integers, and in this case the two notions of torsion coincide.

Suppose that *R* is a (commutative) principal ideal domain and *M* is a finitely-generated *R*-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module *M* up to isomorphism. In particular, it claims that

This page was last edited on 2 July 2017, at 17:30.

Reference: https://en.wikipedia.org/wiki/Torsion_submodule under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Torsion_submodule under CC BY-SA license.

- Abstract Algebra
- Groups
- Modules
- Ring
- Module
- Ring
- Regular Element
- Zero Divisor
- Integral Domain
- Commutative Ring
- Torsion-free
- Lam 2007
- Ore Ring
- Noetherian
- Domain
- Group
- Order
- Integer
- Identity Element
- Torsion (or Periodic) Group
- Abelian Group
- Integers
- Principal Ideal Domain
- Finitely-generated R-module
- Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain

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