# Torsion submodule

Let $R$ be an associative ring with unit, and $M$ a left $R$- module. The torsion subgroup $T( M)$ is defined as

$$T( M) =$$

$$= \ \{ {x \in M } : { \mathop{\rm Ann} _ {R} ( x) \textrm{ contains a regular element } } \} .$$

Here a regular element $r \in R$ is an element that is not a zero divisor (neither left nor right).

If $R$ is left Ore (cf. below and Associative rings and algebras), then $T( M)$ is a submodule of $M$, called the torsion submodule. A module $M$ is torsion free if $T( M) = \{ 0 \}$. A module is torsion if $T( M) = M$.

Quite generally, a torsion theory for an Abelian category ${\mathcal C}$ is a pair $( {\mathcal T} , {\mathcal F} )$ of subclasses of the objects of ${\mathcal C}$ such that $\mathop{\rm Mor} _ {\mathcal C} ( T, F ) = \{ 0 \}$ for all $T \in {\mathcal T}$, $F \in {\mathcal F}$ and ${\mathcal T}$ and ${\mathcal F}$ are maximal with this property, i.e. if $\mathop{\rm Mor} _ {\mathcal C} ( X, F ) = \{ 0 \}$ for all $F \in {\mathcal F}$, then $X \in {\mathcal T}$, and if $\mathop{\rm Mor} _ {\mathcal C} ( T, Y) = \{ 0 \}$ for all $T \in {\mathcal T}$, then $Y \in {\mathcal F}$.

The torsion submodules and torsion-free submodules of a left Ore ring $R$ from a torsion theory for the category $R \textrm{ - Mod }$ of left $R$- modules.

A radical on $R \textrm{ - Mod }$ is a left-exact functor $\mathop{\rm Rad} : R \textrm{ - Mod } \rightarrow R \textrm{ - Mod }$ such that for all $M, N \in R \textrm{ - Mod }$,

i) $\mathop{\rm Rad} ( M)$ is a submodule of $M$;

ii) $f ( \mathop{\rm Rad} M ) \subset \mathop{\rm Rad} N$ for all $f \in \mathop{\rm Hom} _ {R} ( M, N)$; more precisely, $\mathop{\rm Rad} ( f )$ is the restriction of $f$ to $\mathop{\rm Rad} ( M) \subset M$.

iii) $\mathop{\rm Rad} ( M / \mathop{\rm Rad} ( M) ) = \{ 0 \}$.

A radical $\mathop{\rm Rad}$ is a torsion radical or hereditary radical if $N \cap \mathop{\rm Rad} ( M) = \mathop{\rm Rad} ( N)$ for each submodule $N$ of a module $M$. A torsion radical $\mathop{\rm Rad}$ defines a torsion theory for $R \textrm{ - Mod }$ with ${\mathcal T} _ { \mathop{\rm Rad} } = \{ {M \in R \textrm{ - Mod } } : { \mathop{\rm Rad} ( M) = M } \}$, ${\mathcal F} _ { \mathop{\rm Rad} } = \{ {M \in R \textrm{ - Mod } } : { \mathop{\rm Rad} ( M) = 0 } \}$. All torsion theories for $R \textrm{ - Mod }$ arise this way.

A left denominator set of $R$ is a submonoid $S$ of $R$( i.e. $1 \in S$ and $s _ {1} , s _ {2} \in S \Rightarrow s _ { 1 _ 2 } \in S$) such that:

a) (the left Ore condition) for all $s _ {1} \in S$, $r _ {1} \in R$ there are $s _ {2} \in S$, $r _ {2} \in R$ such that $s _ {2} r _ {1} = r _ {2} s _ {1}$;

b) if $r s = 0$ for $r \in R$, $s \in S$, then there is an $s ^ \prime \in S$ with $s ^ \prime r = 0$.

If the set of all regular elements of $R$ is a left denominator set, then $R$ is called left Ore. A left denominator set is also called a left Ore set.

A left denominator set $S$ defines a torsion theory $( {\mathcal T} _ {s} , {\mathcal F} _ {s} )$ for $R \textrm{ - Mod }$ by the associated radical functor

$$\mathop{\rm Rad} _ {S} ( M ) = \ \{ {x \in M } : { \mathop{\rm Ann} _ {R} ( x) \cap S \neq \emptyset } \} .$$

This illustrates the links between torsion theories and (non-commutative) localization (theories). For much more about this theme cf. [a1], [a2], [a3], cf. also Fractions, ring of.

#### References

 [a1] L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) pp. §3.4 [a2] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) pp. §15, §16 [a3] J.S. Golan, "Localization of noncommutative rings" , M. Dekker (1975)
How to Cite This Entry:
Torsion submodule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_submodule&oldid=48999