This question is on a sample exam and I'm struggling.
Let $R=M_3(\mathbb{R})$ be the ring of 3x3 real matrices. Determine a composition series of $R$ as
(1) an $R$-module
and
(2) an $\mathbb{R}$-module.
So, in each problem, we need a strictly decreasing series of submodules of $R$ such that the factors are simple.
My wording is probably going to be off, because I'm new to the subject. My apologies.
In the first problem, we consider $R$ as an $R$ module. This doesn't give us any additional structure on $R$, so we can look at this as the problem of finding a decreasing sequence of subrings such that each subsequent subring is a maximal ideal in the previous. Then the factors would be simple and we would have ourselves a compositions series of $R$ as an $R$-module.
Do the definitions of "simple" for a ring and module coincide like this?
I don't really know how to proceed with this. Something about $R$ being matrices over $\mathbb{R}$ rather than $\mathbb{Z}$ is making it hard for me to find submodules/subring/ideals.
Any suggestions?
Thank you.
Edit:
I posted this without having looked at the second question. I see now that $R$ as an $\mathbb{R}$-module is a vector space, because $R$ is a field. Thus, it has a basis $\mathcal{B}=\{b_1,b_2,...,b_9\}$ and we can probably create a decreasing series of groups each of which is generated by some subset of the basis elements, a la
$ R \supset
I think the following is true:
Each quotient is isomorphic to $