How to compute the Jacobson radical for a matrix ring over $ p$-adic fields?

Question

I met a problem in dealing with representations of $p$-adic groups. The problem is to compute the Jacobson radical of such a matrix ring:

$$ \left(\begin{array}{ll} \mathcal{O}_F & \mathcal{O}_F\\ \mathfrak{P}_F & \mathcal{O}_F \end{array}\right) $$ where the notations are used in the usual sense: $F$ is a non-archimedean local field, $\mathcal{O}_F$ its ring of integers, and $\mathfrak{P}_F$ the maximal ideal.

I know the example of the full matrix ring: if $R$ is a ring with Jacobson radical $\mathfrak{J}$, then the Jacobson radical of the full matrix ring $M_n(R)$ is exactly $M_n(\mathfrak{J})$. (I learnt this example from Lam's famous GTM text, A First Course on Noncommutative Rings.) But I have no idea how to deal with the above "variation". (I have no experience in dealing with hereditary orders or maximal orders of simple algebras over local fields: maybe I should learn the monograph Maximal Orders by Reiner? )

Any hint or help would be welcome. Thanks a lot in advance!

Answer 1

Interesting question. While researching it I ran across Sands, A. D. "Radicals and Morita contexts." Journal of Algebra 24.2 (1973): 335-345. which has the following nice theorem:

Viewing your ring as a Morita context ring $(R, V, W, S)=(\mathcal O_F, \mathcal O_F, \mathfrak P_F, \mathcal O_F)$.

Elsewhere in the paper they do confirm that the Jacobson radical is an $N$-radical, and rewriting this in terms of your rings it'd be

$$\begin{bmatrix}\mathfrak P_F & \mathcal O_F\\ \mathfrak P_F& \mathfrak P_F\end{bmatrix}$$

Answer 2

Not quite as general as rschwieb's answer, but here's how I would have done it: The units of the ring $$ \left(\begin{array}{ll} \mathcal{O}_F & \mathcal{O}_F\\ \mathfrak{P}_F & \mathcal{O}_F \end{array}\right) $$ are exactly the elements of $$ \left(\begin{array}{ll} \mathcal{O}_F^\times & \mathcal{O}_F\\ \mathfrak{P}_F & \mathcal{O}_F^\times \end{array}\right) $$ (hint: determinant, and formula for inverse of $2 \times 2$-matrices).

Now remember one of the characterisations of the Jacobson radical of a unital ring $R$ is that it consists of all $x$ such $1+RxR \subseteq R^\times$. It is rather clear from this that the two-sided ideal

$$ I :=\left(\begin{array}{ll} \mathfrak{P}_F & \mathcal{O}_F\\ \mathfrak{P}_F & \mathfrak{P}_F \end{array}\right) $$

is contained in $Jac(R)$. To see the converse inclusion $Jac(R) \subseteq I$, consider $R/I$.

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Another way to get there: If one sees $p$-adic matrix rings like that, it is always a good idea to consider their quotient "mod $p$", or here, modulo the ideal

$$ Q :=\left(\begin{array}{ll} \mathfrak{P}_F & \mathfrak{P}_F\\ \mathfrak{P}_F & \mathfrak{P}_F \end{array}\right). $$

You get $R/Q \simeq \left(\begin{array}{ll} k_F & k_F\\ 0& k_F \end{array}\right)$, the upper triangular matrices with entries from the residue field $k_F$. This quotient has Jacobson radical $Jac(R/Q)= \left(\begin{array}{ll} 0& k_F\\ 0&0 \end{array}\right)$, so one hopes one gets $Jac(R)$ by lifting this. (That's not a full proof, but an extra idea how to come up with the "candidate" $$ I :=\left(\begin{array}{ll} \mathfrak{P}_F & \mathcal{O}_F\\ \mathfrak{P}_F & \mathfrak{P}_F \end{array}\right) ). $$