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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?

user26857
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CoffeeCCD
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  • You just have to pick something likely and check. It isn't hard at all to find an example that way. – rschwieb Jun 26 '17 at 16:22
  • A generalization of this example would be: let $W$ be a linear subspace of $k^n$. Then the matrices of $M_n(k)$ such that every column is in $W$ is a right ideal of $M_n(k)$. (So in rschwieb's example, $W$ is the span of $(1, 1)$. And in your example, $W$ is the subspace of vectors with some entries fixed to 0.) – Daniel Schepler Jun 26 '17 at 16:42
  • In fact, I've convinced myself every right ideal of $M_n(k)$ is of this form for some subspace $W$ - and the proof is such that it could reasonably be a problem in an upper-division undergraduate course. If you're interested, you could try to work out a proof, and then post another question if you run into trouble. – Daniel Schepler Jun 26 '17 at 16:56

2 Answers2

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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.)

rschwieb
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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).

Daniel Schepler
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