Socle (Mathematics)
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, the term socle has several related meanings. In the context of a module M over a ring R, the socle of M is the sum of the minimal non-trivial submodules of M. It is denoted Soc(M). In particular, a module is semisimple if and only if Soc(M) = M. So the socle of a module could also be defined as the unique maximal semi-simple submodule. If R is a finite dimensional unital algebra and M a finitely generated R-module then the socle consists precisely of the elements annihilated by the radical o...
High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, the term socle has several related meanings. In the context of a module M over a ring R, the socle of M is the sum of the minimal non-trivial submodules of M. It is denoted Soc(M). In particular, a module is semisimple if and only if Soc(M) = M. So the socle of a module could also be defined as the unique maximal semi-simple submodule. If R is a finite dimensional unital algebra and M a finitely generated R-module then the socle consists precisely of the elements annihilated by the radical of R. In the context of group theory, the socle of a group G, denoted Soc(G), is the subgroup generated by the minimal non-trivial normal subgroups of G. The socle is a direct product of minimal normal subgroups.
