View source for Socle

''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]]s (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====
<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>

====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====
<table>
<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>
</table>

{{TEX|done}}