If R is a commutative ring with unity and not an integral domain and F is a free Rmodule 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.

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 AtiyahMacDonald solution manual(http://dangtuanhiep.files.wordpress.com/2008/09/papaioannoua_solutions_to_atiyah.pdf) 

