Find all left ideals of $M_n(\mathbb{C})$

Question

I am taking an introductory abstract algebra class, closely following Gregory T. Lee's Abstract Algebra // An Introductory Course.

I have been asked to find all left, right, and two-sided ideals of the ring of $n \times n$ complex matrices, but we have not been properly introduced to ideals yet. Doing some reading ahead in the textbook (chapter 9), I think I have a decent understanding of what an ideal is but still no clue how to approach this question.

So far, I plan to use the following theorem from the textbook to state that any two-sided ideal must be the entire ring $M_n(\mathbb{C})$ if it contains an invertible matrix, so I can focus my attention on matrices with determinant $0$.

Theorem 9.2. Let $R$ be a ring with identity. If an ideal $I$ of $R$ contains a unit, then $I = R$.

Next, I wanted to use something from this other question, but I'm afraid the textbook doesn't cover anything like this, e.g. vector spaces, until chapter 12, so I would be hesitant to use it in my solution.

There's also this question which seems to go into Morita equivalence which strays even further from the material in our class.

This Wipedia page had some insight which I thought seemed helpful, but again, seems to reference some linear algebra concepts we haven't learned yet.

When we view $M_n(\mathbb{C})$ as the ring of linear endomorphisms of $\mathbb{C}^n$, those matrices which vanish on a given subspace $V$ form a left ideal. Conversely, for a given left ideal $I$ of $M_n(\mathbb{C})$ the intersection of null spaces of all matrices in $I$ gives a subspace of $\mathbb{C}^n$. Under this construction, the left ideals of $M_n(\mathbb{C})$ are in bijection with the subspaces of $\mathbb{C}^n$.

The textbook itself delves into principal ideals, factor (quotient) rings, ring homomorphisms, and eventually the ring isomorphism theorems in chapter 9, all of which I feel could be suitable to answer this question, but it's really not clear to me how one would go about it.

Are there any ways to answer this question without comparing matrices in the ideals as vectors

Answer 1

I am not sure what kind of proof you are expecting. Here is an elementary one.

Suppose the left ideal (denoted by $\mathcal I$ in the sequel) is not the whole matrix space. As pointed out in a theorem that you have mentioned, the matrices in the ideal are necessarily non-units, i.e., they are singular. Let $A$ be the member of the ideal with the maximum possible rank, say, some $m<n$. If $m=0$, the ideal is clearly zero. Let us suppose that $0<m<n$.

By using elementary row and column operations (or Smith normal form, or rank factorisation), we may write $A=P(I_m\oplus 0)Q$ for some invertible matrices $P$ and $Q$. For any $1\le j\le m$, let $T$ be the permutation matrix corresponding to the transposition $(j,n)$. Then $B=(PTP^{-1})A\in\mathcal I$, and so is $A-B=P(e_j-e_n)e_j^TQ$, where $\{e_1,e_2,\ldots,e_n\}$ is the standard basis of $\mathbb C^n$. For any $1\le i\le n$, take any invertible matrix $R\in M_n(\mathbb C)$ such that $R(e_j-e_n)=e_i$. Then $(PRP^{-1})(A-B)=Pe_ie_j^TQ\in\mathcal I$.

It follows that $W=\operatorname{span}\{Pe_ie_j^TQ: 1\le i\le n,\ 1\le j\le m\}\subseteq\mathcal I$. In turn, we must have $W=\mathcal I$, because for every $X\in\mathcal I\setminus W$, some member of the coset $X+V$ will have rank $>m$.

Note that $W$ can also be written as $\{M\in M_n(\mathbb C):MV=0\}$ where $V$ denotes the subspace $\operatorname{span}\{Q^{-1}e_j: j>m\}$ of $\mathbb C^n$. (This is also true when $m=0$ or $m=n$: just take $V=M_n(\mathbb C)$ or $V=0$ respectively.) This is what “those matrices that vanish on a given subspace $V$” in your cited passage means.