Length of a matrix module
Let $R$ be a finite principal ideal ring and $M \subseteq R^m$ an $R$-module. Let $A$ denote the $R$-module of all $m \times n$ matrices over $R$, with $m \leq n,$ whose column space (the $R$-module generated by their columns) is a submodule of $M$. Then I would like to prove the claim that $\lambda_R(A) = n\lambda_R(M)$, where $\lambda_R$ denotes length as an $R$-module.
In the case that $M$ is free (say of rank $k$), we can argue by considering a basis $B= \{m_1, \dots, m_k\}$ of $M$ and prove the claim by showing that the matrices $M_{ij} = [0 \mid 0 \mid \cdots \mid m_j \mid \cdots \mid 0]$,
whose $i$-th column is $m_j$ and all other columns are $0$, form a basis of $A$, by using the linear independence and spanning properties of $B$ to imply the same properties for the set $\{M_{ij}\}.$ However, I can't see how to extend this argument to the case that $M$ is not free: even if we consider a good enough generating set $B$ of $M$, we cannot use the linear independence to imply the linear independence of $\{M_{ij}\}$.
Is the claim wrong when $M$ is not free? And if it's true, how could we prove it?
Answer
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.

- answered
- 160 views
- $35.00
Related Questions
- Compounding interest of principal P, where a compounding withdrawal amount W get withdrawn from P before each compounding of P.
- How to adjust for an additional variable.
- The last six digits of the number $30001^{18} $
- Evaluate $\int_0^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx$
- ALGEBRA WORD PROBLEM - Trajectory of a NASA rocket
- Equation from Test
- Homomorphism
- Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix} $ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$