Prove that the right $M_n(R)$ submodules of $R_n$ are precisely of the form $J_n$ where $J$ runs through all the right ideals of $R$

Question

Prove that the right $M_n(R)$ submodules of $R_n$ are precisely of the form $J_n$ where $J$ runs through all the right ideals of $R$

So, I'm a bit confused by the statement of this question. is $R_n$ assumed to be $n$-tuples filled with elements from $R$?

Answer 1

is $R_n$ assumed to be $n$-tuples filled with elements from $R$?

Yes, just let the matrices operate on the right of the row vectors by regular matrix multiplication.

It’s not hard to show a subset of that form is a right submodule. To do the other direction, let the collection of elements appearing in the first position be $T$. Prove this is a right ideal of $R$, and then show that each collection of elements on all other fixed positions of $R_n$ are precisely $T$.