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I am trying to find at least one example of rings for every combination of the following properties enter image description here

Any ideas for the remaining cases?

Thank you

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For a non-unital integral ring, take $2\mathbb{Z}$ for example. – Tobias Kildetoft Dec 10 '13 at 19:57
For non-commutative unital integral ring, would $\mathbb{Z}[x,y]$ with $xy = -yx$ be ok ? – M.G Dec 10 '13 at 20:05
You can just take the free $\mathbb{Z}$-algebra on two generators. The notation $\mathbb{Z}[x,y]$ usually means $x$ and $y$ commute, so your proposed ring would be trivial. – Tobias Kildetoft Dec 10 '13 at 20:07
For non-unital, non-integral, commutative, you can take $2\mathbb{Z}/4\mathbb{Z}$. – Tobias Kildetoft Dec 10 '13 at 20:07
Very many thanks – M.G Dec 10 '13 at 20:08
up vote 1 down vote accepted

I see three rows of ???? that you are apparently asking about.

For the first one occurring, you could use the ideal $(x)/(x^3)$ in $F_2[x]/(x^3)$. It is properly inside a local ring, so it doesn't have any idempotents aside from $0$, so no identity is there. It's not a domain since $x^3=0$.

For the second one, the integer quaternions (the subring of the quaternions generated by $i,j,k$ and the integers) (or also the Hurwitz quaternions) will do. They're a domain with identity that isn't commutative.

For the third one, you can take any nontrivial ideal of the integer quaternions.

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