# Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS

I would like to know if somewhere the number of non-isomorphic rings with additive group $$(\mathbb{Z}_{16})^2$$ is mentioned. If not, is someone able to calculate it?

And (easier) the commutative case? A341548(16) in OEIS.

• For other cases we have information at web.archive.org/web/20061002201537/http://… Commented Apr 6, 2021 at 10:00
• What do you mean by a ring? Is it associative with identity? Or not necessarily associative? Or not necessarily with identity? Commented Apr 7, 2021 at 20:11
• simple.wikipedia.org/wiki/Ring_(mathematics) Commented Apr 8, 2021 at 4:31
• @BugsBunny associative Commented Apr 8, 2021 at 4:32
• not necessarily with identity Commented Apr 8, 2021 at 5:04

It is not an answer but rather a method, outlined in https://arxiv.org/abs/1903.01623v1 . We adapt it to the ring $$Z_{16}$$.

We say an algebra is curled if every element $$x$$ of order 16 is linearly dependent with $$x^2$$. An algebra is straight otherwise.

If the algebra is curled, it has a basis $$x,y$$ such that $$x^2=kx$$ and $$y^2=ly$$. WLOG, $$k,l\in\{0,1,2,4,8\}$$. Writing $$xy=ax+by$$ and $$yx=cx+dy$$, we get a collection of 6 elements satisfying equations (3), (4) and (5) from the paper. Any collection will give an algebra. The number of collections is $$5^2\cdot 16^4 = 1638400$$. Such search is within computer possibilities but I suspect that it can be narrowed down much further.

If the algebra is straight, it has a basis $$x,y$$ such that $$x^2=ky$$. Again, we can restrict to $$k\in\{1,2,4,8\}$$. I would choose the smallest such $$k$$ possible among all the bases. If $$k=1$$, then $$xy=yx=bx+cy$$ and $$y^2$$ is determined. In this case, a straight algebra is commutative and is determined by 2 elements: $$16^2 = 258$$ possibilities. The analysis in the paper goes through.

The cases of higher $$k$$ require considerations, not covered in the paper. I suspect they are not difficult with $$k$$ being the smallest possible quite helpful: write $$xy$$ and $$yx$$, making sure that no $$x+ay$$ yields a smaller $$k$$.

To determine the isomorphism types, I would the criterion on page 4 and computer. The group $$GL_2(Z_{16})$$ has 4 generators and $$6\cdot 16^3=393216$$ elements. One can apply the generators to the $$2\times 2$$-matrices determining algebras: this gives all the orbits. There is probably a simpler method of just looking for

• I have to reflect on it. Can you tell me how many of them are commutative? Commented Apr 9, 2021 at 12:01
• I don't really know how rewards work. In this case the answer is not entirely satisfactory. Commented Apr 14, 2021 at 10:28