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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.

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It's true. Do you know german? matheplanet.com/matheplanet/nuke/html/article.php?sid=1168 –  Martin Brandenburg Nov 23 '10 at 9:58
    
Alternatively (if $k$ is finite): mathlinks.ro/Forum/viewtopic.php?t=124137 –  Martin Brandenburg Nov 23 '10 at 10:01
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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)

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