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If R is a commutative ring with unity and not an integral domain and F is a free R-module with rank k,is there a linear independent set with cardinality > k? I prooved that this is not true if R is an integral domain and is true if R is not a commutative ring but i can't see the answer to my question.

edited Nov 23 2010 at 10:22

Michael
1,319●8●15

asked Nov 23 2010 at 9:50

t.k
85●1●6

It's true. Do you know german? matheplanet.com/matheplanet/nuke/html/… – Martin Brandenburg Nov 23 2010 at 9:58

Alternatively (if $k$ is finite): mathlinks.ro/Forum/viewtopic.php?t=124137 – Martin Brandenburg Nov 23 2010 at 10:01

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I think this is a multi-dupe: see mathoverflow.net/questions/30860/… and mathoverflow.net/questions/136/… – Kevin Buzzard Nov 23 2010 at 11:00

t.k · Answer 1 · 2010-12-02 15:16:11Z UTC

Thank you all for your help.The best 2 answers i found are the one of Robin Chapman in this site and the similar one of Papaioannou in the Atiyah-MacDonald solution manual(http://dangtuanhiep.files.wordpress.com/2008/09/papaioannoua_solutions_to_atiyah.pdf)

Cardinality of a linear independent subset of a free module over a commutative ring which is not an integral domain

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