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I came across the term "Socle" of a module defined for a finitely generated module $M$ over a noetherian ring $(A,\mathbb{m})$ as follows

$$\mathrm{Soc}(M) = \lbrace x \in m \mid ax = 0\ \forall a \in \mathbb{m} \rbrace.$$

This definition is same as the definition in wikipedia as the sum of simple modules of $M$. I also read that some properties of a module can be read from the socle of a module which i found interesting. Any references towards possibly the proof of statements here would be greatly appreciated.

Thanks

Stefan4024
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random123
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  • Usually they are to be found among the references from the Wiki page. (I thing the commutative-algebra tag can be replaced by non-commutative algebra as the properties you are looking for are proved for non-commutative tings.) – user26857 Oct 04 '15 at 21:38
  • Replaced the tag following your advice. Usually the references are numbered after the statements and i didnot find any references for any except the last one. – random123 Oct 04 '15 at 22:03
  • @random123 What is a "noetherian ring $(A,m)$"? Do you mean that it is a local ring with maximal ideal $m$? – rschwieb Oct 14 '15 at 18:09
  • @rschwieb Yes. That is what i meant. – random123 Oct 15 '15 at 18:13

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