# Do you know which is the minimal local ring that is not isomorphic to its opposite?

The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.

## 1 Answer

I learned this example from MO-user Johannes Hahn: The algebra is $$A=K/(x^3,y*x,y^2,x^2*y)$$ over a field $$K$$ with 2 elements.

Then $$A$$ as an $$A$$-module as 20 submodules, but $$A^{op}$$ as an $$A^{op}$$-module has 16 submodules. Thus $$A$$ and $$A^{op}$$ are not isomorphic. This also gives an example where $$A$$ and $$A^{op}$$ are not derived equivalent (since local algebras are derived equivalent iff they are isomorphic). Another argument (that works for any field $$K$$) is that $$\Omega_A^{1}(I)$$ has dimension 5 but $$\Omega_{A^{op}}^{1}(I)$$ has dimension 10 when $$I$$ is the indecomposable injective module.

One might wonder whether a finite local algebra over a finite field is isomorphic to its opposite algebra if and only if the number of submodules of the regular module coincide.

It might also be interesting to see a selfinjective local algebra not isomorphic to its opposite algebra.

• Is this example published in a paper? Commented Nov 15, 2020 at 11:03
• It might also be interesting to see a selfinjective local algebra not isomorphic to its opposite algebra. That does seem tricky, given that their left and right ideal lattices are certainly isomorphic in the Artinian case. Commented Feb 22, 2022 at 15:29
• @rschwieb I still do not know an example (or I dont remember), so I asked this here: mathoverflow.net/questions/416761/…
– Mare
Commented Feb 22, 2022 at 15:57