Question 1099290
.
I am not sure what your symbol   M2(IR)  exactly denotes.


If it is  {{{M[2](R)}}},  the ring of  2x2-matrices over the field of real numbers,  then the answer is

<pre>
    There are no non-trivial two-sided ideals in this ring.
</pre>


It is well known fact.


For relevant discussion, &nbsp;see the link

<A HREF=https://math.stackexchange.com/questions/22629/why-is-the-ring-of-matrices-over-a-field-simple>https://math.stackexchange.com/questions/22629/why-is-the-ring-of-matrices-over-a-field-simple</A>


https://math.stackexchange.com/questions/22629/why-is-the-ring-of-matrices-over-a-field-simple