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I'm looking for matrix rings with only one maximal ideal. Or maybe with a few maximal ideals, the thing is that I've noticed that these type of rings allow me to find simple modules $R/I$ where is a maximal ideal and check out if some other properties hold or not. I'm already aware of the fact stated in this question

Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$

However, I'm looking forward to more down to earth examples that allow me to easily find the desired and not desired properties just by looking at their lattices. For example, I worked with the ring

$\left( \begin{matrix} \mathbb{Z}_{2}& \mathbb{Z}_{2}\\ 0& \mathbb{Z}_{2}\\ \end{matrix} \right)$

But I'm looking for more examples like this and with fewer ideals if possible. Any help will be appreciated, particularly if you could explain the process of coming with these examples. Thanks

J. W. Tanner
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Cos
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1 Answers1

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It looks like by "matrix ring" you just mean "ring with elements that are matrices" rather than something of the form $M_n(R)$. The ideals of $M_n(R)$ are all completely determined (by the link you gave and other similar posts about one-sided ideals.)

In that case, then every local finite dimensional algebra over a field is fair game.

For example, you might pick $F[x]/(x^2)$, which is isomorphic to the matrix ring $\left\{\begin{bmatrix}a&b\\0&a\end{bmatrix}\,\middle|\, a,b\in F\right\}$. This is a ring with precisely three (left/right/twosided) ideals.

Another one would be the usual representation of $\mathbb C$ as a subring of $M_2(\mathbb R)$, or you could do the representation of $\mathbb H$ as a subring of $M_2(\mathbb C)$, or of $M_4(\mathbb R)$. Those are all division rings so they all have precisely two (left/right/twosided) ideals.

Try this one yourself: $\mathbb Q[x,y]/(x,y)^2$. Find a matrix representation in terms of 3-by-3 matrices.

If you want something with more than one maximal ideal, you can expand your range to commutative, finite dimensional semilocal algebras. Or perhaps even more simply, a product of $n$ fields, which necessarily has $n$ maximal ideals.

rschwieb
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