# Bounding the length of an R-module of matrices

Loosely related to this: Bounding the length in a module of evaluated skew polynomials

Let $$C$$ be an $$\mathbb{F}_q$$-vector subspace of $$m \times n$$ matrices over $$\mathbb{F}_q$$. Assume WLOG that $$m \geq n$$ and let $$d$$ be the minimum rank of a nonzero matrix in $$C$$, then we have the well-known Singleton bound

$$\operatorname{dim}_{\mathbb{F}_q}(C) \leq m(n-d+1),$$

which also immediately gives an upper bound for the cardinality of $$C$$. I am trying to generalise this inequality to matrices over general finite principal ideal rings and replace the minimum rank $$d$$ with the minimum length (over $$R$$) of the module generated by the rows of a matrix. Now we can obtain an upper bound about the cardinality of $$C$$ in the latter case (see e.g. Singleton Bounds for Codes over Finite Rings,Theorem 1), but I am currently stuck with trying to obtain an upper bound for the length (instead of the cardinality) of $$C$$ as an $$R$$-module, depending on $$m,n,d$$ and $$R$$.

Is there an existing result related to this? I have found precisely $$0$$ articles or books mentioning any results about the length of the row space of a matrix over a ring. Is the situation maybe clearer when $$R$$ is local?