I was thinking about the matrix ring $M_n(\mathbb{F})$ (set of all $n\times n$ matrices over the field $\mathbb{F}$) and I determined the fact that every matrix in this ring is either a unit or a zero divisor. However, I think in general this would not imply that the only ideals of $M_n(\mathbb{F})$ are $M_n(\mathbb{F})$  and $\{0\}$. Does knowing that the fact that I proved help determine what the ideals of $M_n(\mathbb{F})$ are?  I of course know other facts which are that the only ideals of the field $\mathbb{F}$ are $0$ and $\mathbb{F}$ and that if $I$ is an ideal in $\mathbb{F}$ then $M_n(I)$ is an ideal of $M_n(\mathbb{F})$.  Any help is appreciated, thank you.
Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$.  Then why does this ring not contain any two-sided ideal?    Thanks for any clarification, and this is an exercise from the notes of Commutative Algebra by Pete L Clark, of which I thought as simple but I cannot figure it out now.
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