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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 22nd 2012

    I did some editing over at free module, under the section on submodules of free modules. I don’t have Rotman’s book before me, so I can’t check whether he assumes the commutativity hypothesis for proposition 2, but I put it in to be safe. (Actually, I’ll bet it’s needed, since we have to be careful around invariant basis number which holds for commutative rings.) The proof that I added does use this hypothesis.

    Also, I deleted the remark that this is the Nielsen-Schreier theorem in the case R=R = \mathbb{Z}, since NS refers to groups as opposed to abelian groups.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2012

    Thanks. Yeah, I guess I omitted the “commutative” there.

    Concerning NS: the Wikipedia entry makes it sound as if Dedekind’s statement that subgroups of free abelian groups are free is regarded as part of the Nielsen-Schreier theorem. That’s how we got into all this in the first place. But it might not be the right way to look at it, after all.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 22nd 2012
    • (edited Oct 22nd 2012)

    Yeah, I’m not sure. But thanks for alerting me to the WP article; I’ll have a look and think about it some; happy to reinstate the remark if it all seems to fit together.

    Edit: Okay, I see the historical remark in the Wikipedia article, that NS is a non-abelian analogue of Dedekind’s theorem, so I’ll go ahead and reinstate something (edit: written before I saw your last comment, so scratch that). It would indeed be nice to derive this result as a corollary of NS. The example that worried me when I commented earlier on p 1(H)p^{-1}(H) is where we take SS to be a two-element set, with p:F(S)F(S) ab=F(S)/[F(S),F(S)]=xyp: F(S) \to F(S)^{ab} = F(S)/[F(S), F(S)] = \mathbb{Z}x \oplus \mathbb{Z}y the projection, and with HH the subgroup generated by x+yx+y. It looked as though the inverse image p 1(H)p^{-1}(H) might be infinitely generated; at least I didn’t see any relations between elements of the form

    y mxy mx nyx ny^m x y^{-m} x^n y x^{-n}

    all of which map to x+yx+y.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2012

    happy to reinstate the remark if it all seems to fit together.

    Ah, no need to do that. I tried to take care to crosslink the NS entry with the pid-entry, that should be sufficient.

    Thanks for all your work on this. Very much appreciated.