This is a curiosity question that came out of teaching abstract algebra.


Let $F$ be a field, and $n>1$ an integer.


Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices $\begin{pmatrix} a\_{1,1} & a\_{1,2} & \cdots & a\_{1,n} \\ 0 & a\_{2,2} & \cdots & a\_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a\_{n,n} \end{pmatrix}$ with entries in $F$.


Let $J$ be the subset of $F^{n \leq n}$ that consists of all such matrices that satisfy $a\_{i,j} = 0$ for all $\left(i,j\right) \neq \left(1,n\right)$. In other words, $J$ is the set of all $n\times n$-matrices whose only nonzero entry (if any) is in the northeasternmost corner. It is easy to see that $J$ is an ideal of $F^{n \leq n}$. Thus, a quotient $F$-algebra $F^{n \leq n} / J$ is defined (and its elements can be thought of as upper-triangular matrices whose northeasternmost entry is undetermined).



> 
> **Question.** What is the smallest $m$ such that there is an injective $F$-algebra homomorphism from $F^{n\leq n} / J$ to the matrix ring $F^{m\times m}$? In other words, what is the smallest dimension of a faithful representation of the $F$-algebra $F^{n \times n} / J$?
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My suspicion is that it is $2n-2$. Indeed, it is certainly $\leq 2n-2$, since there is an injective $F$-algebra homomorphism $F^{n\leq n} / J \to F^{\left(2n-2\right)\leq \left(2n-2\right)}$ that sends the residue class of a matrix $\begin{pmatrix} a\_{1,1} & a\_{1,2} & \cdots & a\_{1,n} \\ 0 & a\_{2,2} & \cdots & a\_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a\_{n,n} \end{pmatrix}$ to $\begin{pmatrix} a\_{1,1} & a\_{1,2} & \cdots & a\_{1,n-1} & 0 & 0 & \cdots & 0 \\ 0 & a\_{2,2} & \cdots & a\_{2,n-1} & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a\_{n-1,n-1} & 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 & a\_{2,2} & a\_{2,3} & \cdots & a\_{2,n} \\ 0 & 0 & \cdots & 0 & 0 & a\_{3,3} & \cdots & a\_{3,n} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & a\_{n,n} \end{pmatrix}$.


For $n = 3$, this is an embedding $F^{3 \leq 3} / J \to F^{4 \leq 4}$, and I think I have convinced myself by a long and messy argument that no embedding $F^{3 \leq 3} / J \to F^{3 \leq 3}$ exists, but this doesn't rule out an embedding $F^{3 \leq 3} / J \to F^{3 \times 3}$ into arbitrary (rather than triangular) matrices.


The whole thing originated in [my attempts to illustrate the variety (common-sense meaning) of quotient rings](https://www.cip.ifi.lmu.de/%7Egrinberg/t/23wa/23wa.pdf). Specifically, I was trying to show that not every quotient ring is just a subring in disguise. The easiest example for this is $\mathbb{Z} / n\mathbb{Z}$, but I was looking for something less obvious.



Smallest faithful representation of an upper-triangular matrix quotient
Here's an elementary proof that $2n-2$ is a lower bound.


Suppose that
$$V\_1\xrightarrow{\alpha\_1}V\_2\xrightarrow{\alpha\_2}\dots\xrightarrow{\alpha\_{n-2}}V\_{n-1}\xrightarrow{\alpha\_{n-1}}V\_n$$
is a representation of the linearly ordered $A\_n$ quiver $Q$ that is a representation of $FQ/I$, where $I$ is the one-dimensional ideal of the path algebra $FQ$ generated by the longest path, so that $\alpha\_{n-1}\dots\alpha\_1=0$, and that it is faithful as a representation of $FQ/I$.


By faithfulness, $\alpha\_{n-2}\dots\alpha\_1(v\_1)\neq0$ for some $v\_1\in V\_1$. For $1<i<n$ let $v\_i=\alpha\_{i-1}\dots\alpha\_1(v\_1)\in V\_i$.


Also by faithfulness, $\alpha\_{n-1}\dots\alpha\_2(w\_2)\neq0$ for some $w\_2\in V\_2$. For $2<i\leq n$ let $w\_i=\alpha\_{i-1}\dots\alpha\_2(w\_2)\in V\_i$.


For $1<i<n$, $v\_i$ and $w\_i$ are linearly independent (consider their images in $V\_{n-1}$ and $V\_n$), so $v\_1,\dots,v\_{n-1},w\_2,\dots,w\_n$ span a ($2n-2$)-dimensional subrepresentation (which is isomorphic to the representation described in the OP).


