# Examples of non-commutative unity rings where $2$ is a non-zero zero divisor

I would like to see some examples of non-commutative unity rings where $$2$$ is a non-zero zero divisor.
My thoughts on this: Since $$2$$ is a zero divisor there will be some $$x$$ in that ring such that $$2x=0$$. This made me think about rings with finite characterstic and I came up with the simple example $$\left\{\begin{pmatrix} a & b \\ 0& c \end{pmatrix}| a, b, c \in \mathbb{Z}_2\right\}$$, but here we have, in fact, that $$2=0$$. To avoid this, we could take $$a, b, c \in \mathbb{Z}_{2n}$$ where $$n\ge2$$ is an integer, but I guess that there should be other nicer examples. I should mention that I want these examples to be as elementary as possible, so I am not really interested in things that require advanced algebraic structures.

• Note that $2\not\in M_2(\Bbb F_2)$. The elements of the ring are $2\times 2$-matrices. Commented Aug 28, 2020 at 8:37
• Consider the noncommutative polynomial ring $\Bbb Z_2[x,y]$. Commented Aug 28, 2020 at 8:38
• @DietrichBurde Yes, by $2$ I meant $2I_2$. Commented Aug 28, 2020 at 8:52
• So by $2$ do you mean $1_R+1_R$ for a general ring? Because $R$ may not contain numbers. Why do you want to consider this? Commented Aug 28, 2020 at 9:37
• @DietrichBurde Yes, that's what I mean. As for why I want to consider this, I need it for a counterexample to a problem I am working on. Commented Aug 28, 2020 at 10:32

Finding examples "nicer" than $$\begin{bmatrix}\mathbb Z_4& \mathbb Z_4\\0&\mathbb Z_4\end{bmatrix}$$ is really kind of grasping at straws. For simplicity and smallness, it's hard to beat. You can, I suppose drop down a little further to $$\begin{bmatrix}\mathbb Z_4& \mathbb 2Z_4\\0&\mathbb Z_4\end{bmatrix}$$.
For example, you could do the free algebra quotient $$\mathbb Z_4\langle x,y\rangle$$, or even a quotient of it, as long as it doesn't make $$xy=yx$$.
Or pick your favorite nonabelian group $$G$$ and form the group ring $$\mathbb Z_4[G]$$.
Or pick your favorite characteristic $$2$$ ring $$R$$ that isn't commutative, and form $$\mathbb Z_4\times R$$.