Right ideals in a matrix ring 
Are there right ideals of $M_n(k)$ other than those of the form of zeroing one or more rows at a time? 

For example you can take first row to be zero and this is a right ideal. My question is are there ideals other than those of this form?
 A: Yes, for example
$$\begin{bmatrix}1&1\\1&1\end{bmatrix}M_2(k)=\left\{\begin{bmatrix}a&b\\a&b \end{bmatrix}\middle|\,a,b\in k\right\}$$
It has nonzero entries in all rows, but it is clearly not the entire ring.
Another counting argument that works for infinite $k$: There are only $2^n$ possibilities for the ideals you're describing, but there are infinitely many right ideals if $k$ is infinite. (Even if $k$ is finite the example above shows that finiteness does not help.)
A: As a generalization of your example, let $W$ be a linear subspace of $k^n$.  Then the set of $n \times n$ matrices such that every column is in $W$, or equivalently, $\{ A \in M_n(k) \, | \, CS(A) \subseteq W \}$, is a right ideal of $M_n(k)$.  (The additivity, and containing 0, are fairly easy; for the right multiplication condition, use the fact that $CS(AB) \subseteq CS(A)$.)
In your example, $W$ is the subspace of vectors with the corresponding elements of the vector equal to 0.  And in rschwieb's example, $W$ is the span of $(1, 1)$.
In fact, it turns out that every right ideal of $M_n(k)$ is of this form for some subspace $W$.  The proof isn't conceptually hard (though it might possibly be cumbersome notation-wise).
