Is there a non trivial ideal for the set of upper triangular matrices?

the zero matrix is a trivial ideal. Also, the set of upper triangular matrices is an ideal. Are there any other ideals?

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    $\begingroup$ Yes, there are nontrivial ideals in the algebra of upper-triangular matrices. Hint: try to impose some restrictions on the diagonal. $\endgroup$ – Julien Nov 5 '13 at 3:31
  • $\begingroup$ Or, for instance, what about matrices that are zero everywhere except the top-right corner? Between this example and julien's, I think you will be able to find a large family of ideals. $\endgroup$ – user7530 Nov 5 '13 at 3:32

Below I categorize all such nontrivial ideals. It should be clear from the proof how one could easily generate a bunch of nontrivial ideals.

Let $\mathfrak{i}$ be an ideal of $\mathfrak{R}$ and let $\mathbf{T}_{ij}$ be the matrix with $1$ at the $(i,j)$ position and $0$ everywhere else. Take $\mathbf{A}\in \mathfrak{i}$. Then $$\begin{eqnarray*}a_{ij}\mathbf{T}_{ij}&=\mathbf{T}_{ii}\mathbf{A}\mathbf{T}_{jj}&\in \mathfrak{i}\end{eqnarray*}$$ Therefore the set of all $(i,j)$ entries of matrices in $\mathfrak{i}$ (which I will write $I_{ij}$) is an ideal of $\mathbb{Z}$, so we have that $I_{ij}=\left(n_{ij}\right)$ for some $\left(n_{ij}\right)\in \mathbb{Z}$. It follows that $$\mathfrak{i}=\{\mathbf{A}\in\mathfrak{R}:d_{ij}\mid a_{ij}\}$$ for some $d_{ij}\in\mathbb{Z}$. Using Bezout's identity, we have $d_{ij}\mid d_{ik}$ for all $j>k$ and $d_{ij}\mid d_{\ell j}$ for all $i>\ell$. In other words, each $d_{ij}$ divides the predecessors of the same row and all entries underneath in the same column.


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