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Similarly, give an example of an infinite non-commutative ring that does not have a unity.

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Matrices are good also for the infinite case: consider the $2\times2$ matrices over $\mathbb{Z}$ where all entries are even. –  egreg Dec 4 '13 at 21:38

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up vote 4 down vote accepted

$\textbf{Hint:}$ Think about matrix rings. They are usually non-commutative.

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Let $S$ be a semigroup with the multiplication: $ab=a$ for all $a,b$, and $F_2$ be the 2-element field. The semigroup ring $F_2S$ is a desired example.

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Unless I'm sorely mistaken, it seems like this is pretty commutative. $ab = ba = 0$ –  Stahl Dec 4 '13 at 21:22
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@ Stahl: Yes, I wrote another example. –  Boris Novikov Dec 4 '13 at 21:27

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