A two part question here:
- Suppose that $A, B \in M_n(\mathbb{F})$ are such that $\mathbb{F}^n / \mathbb{F}^n A$ is isomorphic to $\mathbb{F}^n / \mathbb{F}^n B$. How can we show that $A$ and $B$ are equivalent as matrices (i.e. there exist $P, Q \in GL_n(\mathbb{F})$ such that $A = PBQ$)?
- In the case of the ring $C(\mathbb{R})$, equipped with pointwise addition and multiplication, I'm asked to find $f, g \in C(\mathbb{R})$ where $C(\mathbb{R}) / \langle f \rangle$ is isomorphic to $C(\mathbb{R}) / \langle g \rangle$, but there are no $r, s \in U(C(\mathbb{R}))$ where $f = rgs$.
The second question in some ways is similar to the case of $n=1$ in the first question, except that $C(\mathbb{R})$ is not a field. For this reason I'm quite confident that $f$ and/or $g$ will be zero divisors, but so far I have not yet been able to find appropriate $f$ and $g$ where I can explicitly (or implicitly) show that $C(\mathbb{R}) / \langle f \rangle$ is isomorphic to $C(\mathbb{R}) / \langle g \rangle$. I tried taking $f$ and $g$ where $f$ and $g$ were zero divisors and $f(x) = g(x-1)$, but this didn't seem to work well.
For the first part, I've seen how one can demonstrate that if $A = PBQ$ then $\mathbb{F}^n / \mathbb{F}^n A$ is isomorphic to $\mathbb{F}^n / \mathbb{F}^n B$, yet the proof doesn't really give too much insight as to how one would use an isomorphism between $\mathbb{F}^n / \mathbb{F}^n A$ and $\mathbb{F}^n / \mathbb{F}^n B$ to construct $P$ and $Q$.
Any help is very much appreciated, and let me know if I can clarify anything further.