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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? – Martin Brandenburg Nov 23 '10 at 9:58
Alternatively (if $k$ is finite): – Martin Brandenburg Nov 23 '10 at 10:01

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(

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