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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.

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  • $\begingroup$ Note that $2\not\in M_2(\Bbb F_2)$. The elements of the ring are $2\times 2$-matrices. $\endgroup$ Commented Aug 28, 2020 at 8:37
  • $\begingroup$ Consider the noncommutative polynomial ring $\Bbb Z_2[x,y]$. $\endgroup$
    – Wuestenfux
    Commented Aug 28, 2020 at 8:38
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    $\begingroup$ @DietrichBurde Yes, by $2$ I meant $2I_2$. $\endgroup$
    – Math Guy
    Commented Aug 28, 2020 at 8:52
  • $\begingroup$ 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? $\endgroup$ Commented Aug 28, 2020 at 9:37
  • $\begingroup$ @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. $\endgroup$
    – Math Guy
    Commented Aug 28, 2020 at 10:32

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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 $\begin{bmatrix}\mathbb Z_4& \mathbb 2Z_4\\0&\mathbb Z_4\end{bmatrix}$.

Otherwise the obvious strategies are to use polynomial rings with noncommuting variables or group rings, or products.

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 $\mathbb Z_4\times R$.

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