A finite associative ring with non-transitive ideals?

Question

I'm after a ring R which is finite and associative, but with non-transitive ideals. That is, there exists some J an ideal of R, and I an ideal of J, such that I is not an ideal of R... Plenty of examples of non-transitivity in infinite settings, but haven't found one that is finite...

Answer 1

A simple example is $R=\mathbb{F}_4[x]/(x^2)$, $J=(x)$, and $I=\{0,x\}$. You can get many similar examples by taking $J$ to be any ideal in a finite ring such that $J^2=0$ and $I$ to be an additive subgroup of $J$ that is not closed under multiplication by general elements of $R$.

Answer 2

Let $R$ be the ring of upper triangular $3\times 3$ matrices over a finite field, i.e. $$ R=\left\{\begin{pmatrix}a&b&c\\d&e&f\\0&0&g\end{pmatrix}: a,b,c,d,e,f,g\in F\right\}. $$ Let $J$ be the ideal $$ J=\left\{\begin{pmatrix}0&0&a\\0&0&b\\0&0&0\end{pmatrix}: a,b\in F\right\} $$ and let $I$ be $$ I=\left\{\begin{pmatrix}0&0&a\\0&0&0\\0&0&0\end{pmatrix}: a\in F\right\} $$