Torsion (algebra)

In abstract algebra, the term torsion refers to elements of finite order in groups and to elements of modules annihilated by regular elements of a ring.

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 s in S such that s annihilates m, i.e., s m = 0. In particular, one can take for S the set of regular elements of the ring R and recover the definition above.

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