# Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.

Let $$M$$ be an $$m$$-by-$$n$$ matrix, here are three definitions$$^5$$ that we could use for rank:

1. $$rk(M) = \min k$$ such that for matrices $$P$$, and $$Q$$ with $$P$$ of size $$m$$-by-$$k$$ and $$Q$$ of size $$k$$-by-$$n$$ we have $$M = PQ$$.

2. $$rrk_+(M)$$ (3. $$crk_+(M)$$) is the maximum number of linearly independent rows (columns) of $$M$$. Where a set $$S$$ of vectors is linearly independent$$^1$$ if for all $$u \in S$$ there is no non-zero $$k$$ such that $$ku$$ can be written as a linear combination of vectors in $$S - \{u\}$$.

Now, if the elements of $$M$$ come from a field then $$rk(M) = rrk_+(M) = crk_+(M)$$. However, if we work with modules over a (semi-)rings then the story is different.

If we are working over a commutative (semi-)ring with no zero divisors$$^2$$ then how far apart can $$rk$$, $$rrk_+$$, and $$crk_+$$ be? Can this be bounded in terms of easily computable properties of $$M$$ and the (semi-)ring? I am primarily interested in finite $$M$$ and "well-behaved"$$^3$$ (semi-)rings$$^4$$.

### Notes

1. This definition of linear independence is not necessarily equivalent to the 'make the $$0$$-vector' definition for arbitrary (semi-)rings because (semi-)rings lack minus. I added the there is no non-zero $$k$$ part (instead of just the $$k = 1$$ case) after @Manos' answer.

2. I ask for no zero divisors, because it is easy to come up with separations for general rings. For instance, consider $$\begin{pmatrix} 2 & 2 \\ 3 & 3 \end{pmatrix}$$ over $$\mathbb{Z}_6$$ and the obvious generalizations. I am not interested in examples like this because they are in some way an artifact of just the ring, not of the module.

3. By "well-behaved", I mean you can assume all the nice properties you want since I want the issue to come from the module-ness aspect, not just the the (semi-)ring itself. In fact, I will be more than happy with an answer just for finite (semi-)rings (that meet my other restrictions).

4. I know that semirings are much less studied than rings, so I will be happy with an answer for integral domains if the "semi-" part makes the question too awkward.

5. I realize that there are many other definitions of rank and dimension for modules and (semi-)rings. I am interested in these three specific definitions of rank not to generalize fields but because they come up in application and I want to learn how to play with them. I appreciate pointers to other definitions or rank/dimension if you point out how they relate to the definitions I provide.

• Migrating to MO by OP request. Feb 20 '14 at 8:32

Consider the ring of integers $\mathbb{Z}$. Then this is a principal ideal domain. Consider the $1 \times 2$ matrix $M = \left[ 2 \, \, \, 3 \right]$. Then $rk(M)=1=rrk_+(M)$, if we make the convention that any non-zero element of a $\mathbb{Z}$-module is linearly independent, but $crk_+(M)=2$ because $2 (3)$ can not be expressed as a linear combination of $3 (2)$ (maybe you can come up with more general examples using this idea).
Any any case, if you have a (finite) module $M$ over an integral domain $R$, then the rank of $M$ is defined as the dimension of the $K$-vector space $M \otimes_R K$, where $K$ is the field of fractions of the integral domain $R$. This turns out to be a very useful notion.
• Your paragraph 1 example is very much like the one I tried to rule out by excluding zero-divisors, and can be generalized to give arbitrary separation by, for instance, taking the first $n$-primes with product $N = p_1\cdot...\cdot p_n$ and then taking a row matrix $M = [a_1 , ... , a_n]$ where $a_j = P/p_j$. Feb 12 '14 at 22:18