A non commutative ring of order 125.

Question

How can I construct a non commutative ring of order 125?

I think the approach is to take $M_n(F)$, since the ring of matrices over field is always non commutative. But I am not sure how to find $n$ or a specific field $F$.

Answer 1

The set $T$ of upper triangular $2 \times 2$ matrices

$$\begin{pmatrix}a & b\\0 & c\end{pmatrix}$$

with entries in the field $\mathbb{F}_5$ is a noncommutative subring of the ring $M_2(\mathbb{F}_5)$ of $2 \times 2$ matrices.

• $T$ is a (sub)ring as it is closed for addition and multiplication.

• its cardinality is $5^3$ because there are 3 independent choices for the entries, each one being in $(\mathbb{F}_5)^3.$

• the noncommutativity is shown on an example:

$$\text{If} \ A:=\begin{pmatrix}0 & 1\\0 & 0\end{pmatrix} \ \text{and} \ B:=\begin{pmatrix}0 & 1\\0 & 1\end{pmatrix}$$

then: $AB=A$ whereas $BA=0$ (the null matrix).

(thus $A$ and $B$ are "zero-divisors").

Important remark: Let me now cite the Wikipedia article (https://en.wikipedia.org/wiki/Finite_ring):

"If a non-commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper triangular 2 × 2 matrix ring over the Galois field of the prime. The study of rings of order the cube of a prime was further developed in (Raghavendran 1969) and (Gilmer & Mott 1973). Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case".

Final remark: I just found that the problem (and the very same solution I gave !) can be found in a good book "Abstract Algebra Manual. Problems and Solutions" 2nd Edition, Ayman Badawi, Nova ed. (http://www.worldcat.org/title/abstract-algebra-manual-problems-and-solutions/oclc/54674579)