I am reading through a proof where the writer looks at a simple ring $A$ with a nonzero right ideal $M$. At a certain point he comes to this:
The poduct ideal $AM$ is a two sided ideal, and so $A=AM$.
We can say this because we assumed $A$ is simple. However I'm confused about the fact that $AM\subset A\cap M$, since it would seem to me that $A\cap M=M$ implying $M=A$. This last part seems ludicrous to me, since we have made extremely general assumtions?
At what point am I loosing it?
Since $M$ is a right ideal, we have $MA\subset M$, but not $AM\subset M$, so that $AM\not\subset A\cap M=M$.
As a helpful working example, let $A$ be $n\times n$ matrices over a field, and let $M=M_i$ be all matrices with $0$ entries everywhere except in row $i$. You should be able to see that $M$ is a right ideal, and that $AM=A$.