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Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?

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up vote 8 down vote accepted
+100

This book by J. Lambek says on page 168 that the notion of socle (for modules) is due to Dieudonne. In this paper, the word "socle" is used by Dieudonne, and he explains that he follows M.R. Remak, who first introduced the notion of socle for finite groups, as a subgroup generated by all minimal normal subgroups. However, I cannot find yet where the notation "Soc" appears first (in the context of modules).

Hope this bit of information helps. Cheers!

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According to Richard Brauer jstor.org/stable/1989943 the notion of socle (sockel) at least for groups goes back to G. Frobenius, Theorie der hyperkomplexen Grossen, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1903, pp. 504-537. –  Name Oct 5 '13 at 18:19
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Just to establish some more reference points:

In the book
[Curtis, Charles W.; Reiner, Irving. Representation theory of finite groups and associative algebras. Pure and Applied Mathematics, Vol. XI Interscience Publishers, a division of John Wiley & Sons, New York-London 1962],
the socle of a module is defined, but no special notation is used.

In the paper
[Curtis, C. W.; Jans, J. P. On algebras with a finite number of indecomposable modules. Trans. Amer. Math. Soc. 114 1965 122--132.],
the notation $S(M)$ is used.

In the book
[Anderson, Frank W.; Fuller, Kent R. Rings and categories of modules. Graduate Texts in Mathematics, Vol. 13. Springer-Verlag, New York-Heidelberg, 1974],
the notation ${Soc} \; M$ is used.

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