Timeline for Different definitions of the rank of a module
Current License: CC BY-SA 3.0
12 events
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| Jun 22, 2022 at 8:13 | history | edited | CommunityBot |
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| Jan 6, 2014 at 19:49 | vote | accept | brunoh | ||
| Jan 6, 2014 at 19:43 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Jan 6, 2014 at 2:52 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Jan 6, 2014 at 0:25 | comment | added | user9072 | I first replied but deleted the comment upon seeing you already had edited after your comment. Thanks for taking the remark into account. Let me just add that I think still one more notion of rank for abelian groups around namely minimal cardiniality of generating set, so that all cyclic groups have rank 1, and for fin gen case the rank corresponds to the number of cyclic groups one uses when taking as orders a chain of divisors for the torsion part. | |
| Jan 6, 2014 at 0:12 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Jan 6, 2014 at 0:02 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Jan 6, 2014 at 0:00 | comment | added | Pete L. Clark | @quid: What is the definition of rank given in Fuchs? | |
| Jan 5, 2014 at 23:14 | comment | added | user9072 | IMO, the final remarks on abelian groups could be a bit misleading. Namely, they seem quite at odds with the treatment of 'rank like' notions in 'Infinite Abelian Groups' by Fuchs which is AFAIK a standard reference on the subject. First, rank of an abelian group there is defined differently so the 'usually' potentially is an overstatement. Second, the rank function is defined and used, via defining p-rank, for p prime, and torsion-free rank (which would be what is called rank here); rank there would be the sum of all the p-ranks and torsion-free rank. | |
| Jan 5, 2014 at 21:46 | comment | added | brunoh | @PeteL.Clark thank you for your answer. It is clear for the comparison between 1. And 3. But what I was trying to find is a way to make 2. well defined in more general situations : for instance when $M$ is not locally free , one can still look at the maximum number of independents elements locally, right ? The difference between 1. and 2. is then a measure of the failure of local freeness for $M$. | |
| Jan 5, 2014 at 21:19 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Jan 5, 2014 at 20:32 | history | answered | Pete L. Clark | CC BY-SA 3.0 |