Socle

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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.

Contents

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