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 $F^{n \times n} / J$?

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. 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.

2more comments