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Let $R$ be a unital *-ring. Assume that $R$ has finitely many projections.

Q. Can we conclude that $R$ is finite?!

$\bullet$ We say $R$ is finite if $x^*x=1_R$ implies that $xx^*=1_R$.

$\bullet$ $p\in R$ is called a projection if $p=p^*=p^2$.

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    $\begingroup$ Yes, it is easy - for $x^*x=1$, every $x^n(x^*)^n$ is a projection, and if two of them are equal then multiplying both sides of that equality by $x^*$ from the left and by $x$ from the right several times we get $xx^*=1$. $\endgroup$ May 31, 2017 at 7:18
  • $\begingroup$ @მამუკაჯიბლაძე: I agree that you get $x^n (x^*)^n = 1$ for some $n$, but why does it follow that $n=1$? $\endgroup$
    – Alex M.
    May 31, 2017 at 7:34
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    $\begingroup$ @AlexM. just keep multiplying :) $\endgroup$ May 31, 2017 at 8:17

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