There is a statement.
Let $R$ be an integral domain and let $M$ be a free $R$-module of rank $n<\infty$. Then any $n+1$ elements of $M$ are $R$-linearly dependent. (This is extracted from the book 'Abstract Algebra' written by David S. Dummit and Richard M. Foote.)
Let's make the condition weaker as we assume $R$ is commutative ring with unity. Then, what is an example that the number of linearly independent element of $M$ exceeds the rank?
