I was reading Lam's "Lectures on Modules and Rings" and the results I found can be summarized like this:
$$ \text{strong rank condition} \Rightarrow \text{rank condition} $$ and $$\text{rank condition} \Rightarrow \text{IBN}$$
From Lam's book:
- A ring $R$ satisfies the rank condition, if, $\forall n<\infty$ any set of $R$ module generators for $R^n$ has cardinality $\geq n$.
- A ring $R$ satisfies the strong rank condition, if, $\forall n<\infty$, any set of linearly independent elements in $R^n$ has cardinality $\leq n$.
Are there any (other that the ones in Lam's book) intuitive examples that show that the converse is not true for neither of those affirmations?
My approach was to try and find a comutative ring for the second but every known ring I tried satisfies the rank condition as well.
Proceed to pages 11 and 13 of the book you are reading:
Page 11: Let $R=\mathbb Q\langle a,b,c,d\rangle /(ac-1, bd-1, ab, cd)$. Then $R$ does not have the rank condition but it is IBN.
Example 1.31 page 13 (paraphrased): The free algebra $k\langle x,y \rangle$ satisfies the rank condition but not the strong rank condition.
Note: The request for examples other than those in the book was not in the question when this answer was offered.