Question

Let I be an ideal of a ring R contained in rad R (the Jacobson radical of R). Show that R is stably finite iff R/ I is.

Answer 1

We shall use the following crucial fact about the Jacobson radical of a ring $S$:

if $J$ is an ideal in $\operatorname{rad}(S)$ and $\overline{S} = S / J$ then $a \in S$ has an inverse iff $\overline{a} \in \overline{S}$ does.

Then we deduce that $S$ is Dedekind – finite iff $\overline{S}$ is.

Let $S = M_{n}(R)$. Then

$$J = M_{n}(I) \subseteq M_{n}(\operatorname{rad} R) = \operatorname{rad} S$$ $$S / J = M_{n}(R) / M_{n}(I) \cong M_{n}(R / I)$$

Then $S = M_{n}(R)$ is Dedekind – finite iff $S / J = M_{n}(R / I)$ is. Since this holds for all $n$, then $R$ is stably finite iff $R / I$ is.