I found the theorem,
"If I is an ideal of R, then $M_n(I)$ is an ideal of $M_n(R)$"
while trying to work on an example and wonder how to use it prove the following example. I saw another thread to do with this theorem but did not completely understand the concept and how to relate it to my example. Is it something to do with showing that it is closed under addition and multiplication?
The example is:
R is a ring and I is an ideal of R. Show that
$M_2(I) = \left\{ \begin{pmatrix}a & b\\\ c & d\end{pmatrix} : a,b,c,d \in I \right\}$,
is an ideal of
$M_2(R) = \left\{ \begin{pmatrix}a & b\\\ c & d\end{pmatrix} : a,b,c,d \in R \right\}$
Thank you and I hope I have given enough information.
The relationship is this: your example is the case $n=2$ of the "theorem you found."
I'm not sure what "it" is, but I gather that you're asking something like "do I have to show $M_n(I)$ is an ideal because it satisfies the axioms defining ideals," and the answer to that would be "yes."
This fully answers the questions in your post as currently written. If you perhaps realize you wanted to ask something else more specific, you can add to your post and ping me with a comment to add to this solution.