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| − | A [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933201.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933202.png" /> without divisors of zero, such that the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933205.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933206.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933207.png" />. Examples of such (left) modules are the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933208.png" /> itself and all its non-zero left ideals. A submodule of a torsion-free module and also the direct sum and direct product of torsion-free modules are torsion-free modules. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t0933209.png" /> is commutative, then for any module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332010.png" /> there is a torsion submodule | + | {{TEX|done}} |
| | + | A [[Module|module]] $M$ over a ring $A$ without divisors of zero, such that the equality $am=0$, where $a\in A$, $m\in M$, implies $a=0$ or $m=0$. Examples of such (left) modules are the ring $A$ itself and all its non-zero left ideals. A submodule of a torsion-free module and also the direct sum and direct product of torsion-free modules are torsion-free modules. If $A$ is commutative, then for any module $M$ there is a torsion submodule |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332011.png" /></td> </tr></table>
| + | $$T(M)=\{m\in M\colon\exists a\in A,a\neq0,am=0\}.$$ |
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| − | In this case the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332012.png" /> is torsion-free. | + | In this case the quotient module $M/T(M)$ is torsion-free. |
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| | ====Comments==== | | ====Comments==== |
| − | More generally, for any associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332013.png" /> a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332014.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332015.png" /> is called torsion-free if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332017.png" /> for a regular element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332018.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093320/t09332019.png" />. Cf. [[Torsion submodule|Torsion submodule]] for more details and some references. | + | More generally, for any associative ring $R$ a left $R$-module $M$ is called torsion-free if for $m\in M$, $rm=0$ for a regular element $r\in R$ implies $m=0$. Cf. [[Torsion submodule|Torsion submodule]] for more details and some references. |
Latest revision as of 07:40, 22 August 2014
A module $M$ over a ring $A$ without divisors of zero, such that the equality $am=0$, where $a\in A$, $m\in M$, implies $a=0$ or $m=0$. Examples of such (left) modules are the ring $A$ itself and all its non-zero left ideals. A submodule of a torsion-free module and also the direct sum and direct product of torsion-free modules are torsion-free modules. If $A$ is commutative, then for any module $M$ there is a torsion submodule
$$T(M)=\{m\in M\colon\exists a\in A,a\neq0,am=0\}.$$
In this case the quotient module $M/T(M)$ is torsion-free.
More generally, for any associative ring $R$ a left $R$-module $M$ is called torsion-free if for $m\in M$, $rm=0$ for a regular element $r\in R$ implies $m=0$. Cf. Torsion submodule for more details and some references.
How to Cite This Entry:
Torsion-free module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion-free_module&oldid=33072
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article