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''of a module''
 
''of a module''
  
The sum of all its simple submodules. When there are none, the socle is taken to be 0. In accordance with this definition one can consider in a ring its left and right socle. Each of them turns out to be a two-sided ideal that is invariant under all endomorphisms of the ring. The socle can be represented as a direct sum of simple modules. [[Completely-reducible module|Completely-reducible modules]] (semi-simple modules) can be characterized as modules that coincide with their socle.
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The sum of all its simple submodules. When there are none, the socle is taken to be 0. In accordance with this definition one can consider in a ring its left and right socle. Each of them turns out to be a two-sided ideal that is invariant under all endomorphisms of the ring. The socle can be represented as a direct sum of simple modules. [[Completely-reducible module]]s (semi-simple modules) can be characterized as modules that coincide with their socle.
  
  
  
 
====Comments====
 
====Comments====
A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859901.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859902.png" /> is large, or essential, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859903.png" /> for every non-zero submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859904.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859905.png" />. A complement (respectively, essential complement) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859906.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859907.png" /> is a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859908.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s0859909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599010.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599012.png" /> is large). A module is complemented if each submodule has a complement. Each submodule always has a (not necessarily unique) essential complement. A module is complemented if and only if it is completely reducible and hence if and only if it coincides with its socle. The socle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599013.png" /> can also be defined as the intersection of all the essential submodules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599014.png" />. The socle is the largest semi-simple submodule.
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A submodule $N$ of a module $M$ is a large, or [[essential submodule]], if $N \cap N' \ne 0$ for every non-zero submodule $N'$ of $M$. A complement (respectively, essential complement) of $N$ in $M$ is a submodule $N'$ such that $N \cap N' = 0$ and $N + N' = M$ (respectively, $N \cap N' = 0$ and $N + N'$ is large). A module is complemented if each submodule has a complement. Each submodule always has a (not necessarily unique) essential complement. A module is complemented if and only if it is completely reducible and hence if and only if it coincides with its socle. The socle of $M$ can also be defined as the intersection of all the essential submodules of $M$. The socle is the largest semi-simple submodule.
  
More generally, for a modular lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599015.png" /> an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599016.png" /> is large or essential if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599018.png" />. The socle of a modular lattice is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599019.png" />. The interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085990/s08599020.png" /> is a complemented lattice.
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More generally, for a [[modular lattice]] $L$ an element $a \in L$ is large or essential if $a \wedge b \ne 0$ for all $b \ne 0$. The socle of a modular lattice is defined as  
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$$
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\mathrm{soc}(L) = \bigwedge \{a \in L : a\ \text{large}\} \ .
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$$
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The interval $[0,\mathrm{soc}(L)]$ is a complemented lattice.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''1''' , Acad. Press  (1988)  pp. §2.4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)  pp. 367</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''1''' , Acad. Press  (1988)  pp. §2.4</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)  pp. 367</TD></TR>
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</table>
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====Comments====
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The socle of a group is the subgroup generated by the minimal normal subgroups: it is a [[characteristic subgroup]].  It is a direct product of minimal normal subgroups.
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====References====
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<table>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  Derek Robinson, "A Course in the Theory of Groups", Graduate Texts in Mathematics '''80''' Springer (1996)  {{ISBN|0-387-94461-3}}</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 15:04, 19 November 2023

of a module

The sum of all its simple submodules. When there are none, the socle is taken to be 0. In accordance with this definition one can consider in a ring its left and right socle. Each of them turns out to be a two-sided ideal that is invariant under all endomorphisms of the ring. The socle can be represented as a direct sum of simple modules. Completely-reducible modules (semi-simple modules) can be characterized as modules that coincide with their socle.


Comments

A submodule $N$ of a module $M$ is a large, or essential submodule, if $N \cap N' \ne 0$ for every non-zero submodule $N'$ of $M$. A complement (respectively, essential complement) of $N$ in $M$ is a submodule $N'$ such that $N \cap N' = 0$ and $N + N' = M$ (respectively, $N \cap N' = 0$ and $N + N'$ is large). A module is complemented if each submodule has a complement. Each submodule always has a (not necessarily unique) essential complement. A module is complemented if and only if it is completely reducible and hence if and only if it coincides with its socle. The socle of $M$ can also be defined as the intersection of all the essential submodules of $M$. The socle is the largest semi-simple submodule.

More generally, for a modular lattice $L$ an element $a \in L$ is large or essential if $a \wedge b \ne 0$ for all $b \ne 0$. The socle of a modular lattice is defined as $$ \mathrm{soc}(L) = \bigwedge \{a \in L : a\ \text{large}\} \ . $$ The interval $[0,\mathrm{soc}(L)]$ is a complemented lattice.

References

[a1] L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) pp. §2.4
[a2] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) pp. 367

Comments

The socle of a group is the subgroup generated by the minimal normal subgroups: it is a characteristic subgroup. It is a direct product of minimal normal subgroups.

References

[b1] Derek Robinson, "A Course in the Theory of Groups", Graduate Texts in Mathematics 80 Springer (1996) ISBN 0-387-94461-3
How to Cite This Entry:
Socle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Socle&oldid=18319
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article