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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:

  1. A ring $R$ satisfies the rank condition, if, $\forall n<\infty$ any set of $R$ module generators for $R^n$ has cardinality $\geq n$.
  2. 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.

Pingu
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    I think that you would be more likely to get answers if you defined "rank condition" and "strong rank condition". Also, is there some reason that the examples in Lam's book don't satisfy you? – Jeremy Rickard Jun 03 '23 at 18:30
  • I was courious if there are other known examples besides the ones in Lam's book. – Pingu Jun 04 '23 at 07:02
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    @Pingu Certainly there are, but the question you would have to answer in that case is "Why do you need different ones? What if anything should be different?" We can't guess at what you might want. You might say "are there easier examples?" and to that the answer is probably "no" because that author is quite capable of giving the most straightforward examples. – rschwieb Jun 04 '23 at 15:03

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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.

rschwieb
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  • Thank you for the answer! I was wondering if there were other known examples, besides the ones in Lam's book. – Pingu Jun 04 '23 at 07:12
  • The first example should be $R=\mathbb Q\langle a,b,c,d\rangle/(ac-1, bd-1, ab, cd)$ (i.e., the variables don't commute). Every commutative ring satisfies the rank condition. – Jeremy Rickard Jun 05 '23 at 08:16
  • @JeremyRickard Oops yes, forgot to use the right delimiters... thank you. should be good now. – rschwieb Jun 05 '23 at 14:59