Timeline for Why is it a good idea to study a ring by studying its modules?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2015 at 19:00 | comment | added | Daniel McLaury | (Not that I think I'm saying anything original or new to you here; it's just something that I think would be helpful to someone first learning about rings and modules.) | |
| Jul 6, 2015 at 18:58 | comment | added | Daniel McLaury | So in some sense abelian groups are only groups "by accident." I feel that the situation with commutative rings is entirely analogous: an object in the category of rings is naturally thought of as a ring of operators, whereas an object in the category of commutative rings is naturally thought of as a ring of coordinate functions. | |
| Jul 6, 2015 at 18:57 | comment | added | Daniel McLaury | But in some sense, commutative rings are a totally different animal from arbitrary rings, and shouldn't be viewed as a special case. Above, you point out that thinking of groups without considering their actions is the "wrong way of thinking," and I agree -- but notice that this applies primarily to nonabelian groups. Abelian groups tend to arise as groups of formal linear combinations of things, and in this context they don't tend to come with interesting group actions. | |
| Jul 6, 2015 at 18:54 | comment | added | Daniel McLaury | I think this is a good way of thinking about things for $R$ a general ring, but there's one thing I think is off-point. You observe that an arbitrary ring can be realized as a subring of End(A) for some abelian group, just as an arbitrary group can be realized as a subgroup of Sym(S) for some set S, and then say "Note that End(M) is in general noncommutative, so this construction is more general than any "ring of functions" construction in (commutative!) algebraic geometry." | |
| Mar 24, 2012 at 23:36 | comment | added | Dejan Govc | +1. Beautifully written. | |
| Mar 28, 2011 at 19:29 | comment | added | emperordali | The introduction to section 12.A in Isaacs Algebra: A Graduate Course talks a little about this view. | |
| Nov 16, 2009 at 22:17 | comment | added | Jose Brox | Thanks Pete! Could you recommend a book about ring/modules/group-theory with this view as its driving force? I think I am lacking this kind of insight. | |
| Nov 16, 2009 at 21:59 | history | answered | Pete L. Clark | CC BY-SA 2.5 |