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…mehr

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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 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.