Difference between revisions of "Socle"
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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> | <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> | <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> | ||
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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 |
Socle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Socle&oldid=42167